Revision as of 20:09, 13 March 2016

Nonaxisymmetric Instability in Papaloizou-Pringle Tori

Linearized Principal Governing Equations in Cylindrical Coordinates

We begin by drawing from an accompanying derivation the relevant set of linearized principal governing equations, written in cylindrical coordinates but, following the lead of Papaloizou & Pringle (1984, MNRAS, 208, 721-750; hereafter, PP84), express each perturbation in the form,

<math>~q^'~~\rightarrow~~ q^' (\varpi,z) f_\sigma</math>

where,

<math>~f_\sigma \equiv e^{i(m\varphi + \sigma t)} \, ,</math>

and, set <math>~\Phi^' = 0</math> — hence, the Poisson equation becomes irrelevant — because the torus is assumed not to be self-gravitating and the background (point source) potential, <math>~\Phi_0</math>, is assumed to be unchanging.

Set of Linearized Principal Governing Equations in Cylindrical Coordinates

Continuity Equation
<math>~\frac{\partial (\rho^' f_\sigma) }{\partial t} + ( {\dot\varphi}_0 )\frac{\partial (\rho^' f_\sigma)}{\partial \varphi} </math>	<math>~=</math>	<math>~ - \frac{f_\sigma}{\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}^' f_\sigma\biggr] - f_\sigma\frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] </math>
<math>\varpi</math> Component of Euler Equation
<math>~ \frac{\partial ({\dot\varpi}^'f_\sigma) }{\partial t} + ( {\dot\varphi}_0 ) \frac{\partial ( {\dot\varpi}^'f_\sigma)}{\partial\varphi} - 2\varpi ( {\dot\varphi}_0 {\dot\varphi}^' f_\sigma) </math>	<math>~=</math>	<math>~ - f_\sigma\frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) </math>
<math>\varphi</math> Component of Euler Equation
<math>~\frac{\partial (\varpi {\dot\varphi}^' f_\sigma)}{\partial t} + ( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^' f_\sigma)}{\partial\varphi} + \frac{{\dot\varpi}^' f_\sigma}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] </math>	<math>~=</math>	<math>~- \frac{ 1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'f_\sigma}{\rho_0}\biggr) \biggr] </math>
<math>~z</math> Component of Euler Equation
<math>~ \frac{\partial ({\dot{z}}^' f_\sigma)}{\partial t} + (\dot\varphi_0) \frac{\partial ({\dot{z}}^' f_\sigma)}{\partial\varphi} </math>	<math>~=</math>	<math>~ - f_\sigma \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) </math>
Adiabatic Form of the 1^st Law of Thermodynamics
<math>~\frac{P^' f_\sigma}{P_0}</math>	<math>~=</math>	<math>~ \frac{\gamma (\rho^' f_\sigma)}{\rho_0} </math>

Next, taking derivatives of <math>~f_\sigma</math>, where indicated, then dividing every equation through by <math>~f_\sigma</math> gives,

Linearized Adiabatic Form of the 1^st Law of Thermodynamics
<math>~\frac{P^' }{P_0}</math>	<math>~=</math>	<math>~ \frac{\gamma \rho^' }{\rho_0} \, ;</math>
Linearized <math>\varpi</math> Component of Euler Equation
<math>~{\dot\varpi}^'[i(\sigma + m{\dot\varphi}_0)] - 2 {\dot\varphi}_0 (\varpi {\dot\varphi}^' ) </math>	<math>~=</math>	<math>~ - \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \, ; </math>
Linearized <math>\varphi</math> Component of Euler Equation
<math>~(\varpi {\dot\varphi}^')[i(\sigma + m{\dot\varphi}_0)] + \frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] </math>	<math>~=</math>	<math>~- \frac{ im}{\varpi} \biggl(\frac{P^'}{\rho_0}\biggr) \, ; </math>
Linearized <math>~z</math> Component of Euler Equation
<math>~ ~{\dot{z}}^'[i(\sigma + m{\dot\varphi}_0)] </math>	<math>~=</math>	<math>~ - \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) \, ; </math>
Linearized Continuity Equation
<math>~\rho^'[i(\sigma + m{\dot\varphi}_0)] + i m\rho_0 (\varpi {\dot\varphi}^' ) </math>	<math>~=</math>	<math>~ - \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] - \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, . </math>

These five equations match, respectively, equations (3.8) - (3.12) of PP84.

Formulation of Eigenvalue Problem

Again following the lead of PP84, we let <math>~W^'</math> represent the (normalized) perturbation in the fluid entropy, specifically,

<math>~W^' </math>	<math>~\equiv</math>	<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
<math>~\Rightarrow~~~~\frac{\partial}{\partial\varpi}\biggl(\frac{P^'}{\rho_0} \biggr)</math>	<math>~=</math>	<math>~\frac{\partial}{\partial\varpi} \biggl[ W^'(\sigma + m{\dot\varphi}_0 )\biggr]</math>
	<math>~=</math>	<math>~(\sigma + m{\dot\varphi}_0 )\frac{\partial W^'}{\partial\varpi} + mW^'\frac{\partial {\dot\varphi}_0 }{\partial\varpi} </math>

in which case the three linearized components of the Euler equation may be rewritten as,

Linearized <math>\varpi</math> Component of Euler Equation
<math>~{\dot\varpi}^' </math>	<math>~=</math>	<math>~ i \biggl[ \frac{\partial W^'}{\partial\varpi} + \frac{mW^'}{(\sigma + m{\dot\varphi}_0)}\frac{\partial {\dot\varphi}_0 }{\partial\varpi} - \frac{2{\dot\varphi}_0 (\varpi {\dot\varphi}^' )}{(\sigma + m{\dot\varphi}_0)} \biggr] </math>
Linearized <math>\varphi</math> Component of Euler Equation
<math>~(\varpi {\dot\varphi}^') </math>	<math>~=</math>	<math>~- \frac{ mW^'}{\varpi} + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] \, ; </math>
Linearized <math>~z</math> Component of Euler Equation
<math>~ ~{\dot{z}}^' </math>	<math>~=</math>	<math>~ i~\frac{\partial W^'}{\partial z} \, . </math>

Using the second of these three relations to provide an expression for <math>~(\varpi {\dot\varphi}^')</math> in terms of <math>~W^'</math> and <math>~{\dot\varpi}^'</math>, and plugging this expression into the first relation allows us to solve for the radial component of the perturbed velocity in terms of <math>~W^'</math> and its radial derivative. Specifically, we obtain,

<math>~{\dot\varpi}^' </math>	<math>~=</math>	<math>~i \frac{\partial W^'}{\partial \varpi} + i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} - \frac{2 {\dot\varphi}_0 }{\varpi}\biggr] - i~ \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} \biggl[ - \frac{ mW^'}{\varpi} + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr] </math>
	<math>~=</math>	<math>~i \frac{\partial W^'}{\partial \varpi} + i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr] + \biggl[ \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} \biggr]\biggl[ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr] </math>
	<math>~=</math>	<math>~i \biggl[ \frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) \frac{ mW^'}{\bar\sigma} \biggr] + \biggl[ {\dot\varpi}^'\biggl( \frac{ \kappa^2 }{ {\bar\sigma}^2 } \biggr) \biggr] </math>
<math>~\Rightarrow ~~~~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>	<math>~=</math>	<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) mW^' \bar\sigma \biggr] \, , </math>

where, adopting notation from PP84,

<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>

and

<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) \, .</math>

This means, as well, that,

<math>~(\varpi {\dot\varphi}^') ({\bar\sigma}^2 - \kappa^2 ) </math>	<math>~=</math>	<math>~- \frac{ mW^'}{\varpi} ({\bar\sigma}^2 - \kappa^2 ) - \frac{ 1 }{\varpi \bar\sigma }\biggl[ \frac{\kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr] \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) mW^' \bar\sigma \biggr] </math>
	<math>~=</math>	<math>~- \frac{ m{\bar\sigma}^2 W^'}{\varpi} + \frac{ m\kappa^2W^'}{\varpi} - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 } \biggl[ ~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma } \biggr] </math>
	<math>~=</math>	<math>~- \frac{ m{\bar\sigma}^2 W^'}{\varpi} - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 } \biggl[ ~\frac{\partial W^'}{\partial \varpi} +\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma } \biggr] \, . </math>

In summary, the three components of the perturbed velocity are:

Cylindrical-Coordinate Components of the Perturbed Velocity

<math>\varpi</math> Component
<math>~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>	<math>~=</math>	<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) mW^' \bar\sigma \biggr] \, , </math>
<math>\varphi</math> Component
<math>~(\varpi {\dot\varphi}^') ({\bar\sigma}^2 - \kappa^2 ) </math>	<math>~=</math>	<math>~- \frac{ m{\bar\sigma}^2 W^'}{\varpi} - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 } \biggl[ ~\frac{\partial W^'}{\partial \varpi} +\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma } \biggr] \, . </math>
<math>~z</math> Component
<math>~ ~{\dot{z}}^' </math>	<math>~=</math>	<math>~ i~\frac{\partial W^'}{\partial z} \, . </math>

where,

<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>

and

<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) </math>

These three velocity-component expressions match, respectively, equations (3.14), (3.15), and (3.16).

@@ Line 418: / Line 418: @@
    <td align="left">
 <math>~-  \frac{ mW^'}{\varpi} ({\bar\sigma}^2 - \kappa^2 )
-- \frac{ 1 }{\varpi \bar\sigma }\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi}  \biggr]
+- \frac{ 1 }{\varpi \bar\sigma }\biggl[ \frac{\kappa^2 \varpi }{ 2{\dot\varphi}_0 }  \biggr]
 \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi}
-+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr]
++\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) mW^' \bar\sigma   \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  +  \frac{ m\kappa^2W^'}{\varpi}
+- \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi}
++\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]
+</math>
+  </td>
+</tr>
+<tr>
+  <td align="right">
+&nbsp;
+  </td>
+  <td align="center">
+<math>~=</math>
+  </td>
+  <td align="left">
+<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi}
++\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
 </math>
    </td>
@@ Line 428: / Line 457: @@
-Using the first of these three relations to provide an expression for <math>~{\dot\varpi}^'</math> in terms of <math>~W^'</math> and <math>~{\dot\varphi}^'</math>, and plugging this expression into the second relation allows us to solve for the azimuthal component of the perturbed velocity in terms of <math>~W^'</math> and its radial derivative.  Specifically, we obtain,
+In summary, the three components of the perturbed velocity are:
-<div align="center">
-<table border="0" cellpadding="5" align="center">
+<table border="1" cellpadding="8" align="center">
+<tr>
+  <th align="center">
+Cylindrical-Coordinate Components of the Perturbed Velocity
+  </th>
+</tr>
+<tr><td>
+<table border="0" cellpadding="8" align="center">
+<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component'''</font></td></tr>
 <tr>
    <td align="right">
-<math>~(\varpi {\dot\varphi}^')  </math>
+<math>~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>
    </td>
    <td align="center">
@@ Line 440: / Line 479: @@
    </td>
    <td align="left">
-<math>~-  \frac{ mW^'}{\varpi}  - \frac{1}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]
+<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi}
-\biggl[\frac{\partial W^'}{\partial \varpi} -  \frac{2 {\dot\varphi}_0 (\varpi {\dot\varphi}^' )}{(\sigma + m{\dot\varphi}_0)} \biggr]
++\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \, ,
 </math>
    </td>
 </tr>
+<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component'''</font></td></tr>
 <tr>
    <td align="right">
-&nbsp;
+<math>~(\varpi {\dot\varphi}^')  ({\bar\sigma}^2 - \kappa^2 )
+</math>
    </td>
    <td align="center">
@@ Line 454: / Line 495: @@
    </td>
    <td align="left">
-<math>~-  \frac{ mW^'}{\varpi}
+<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi}
-- \frac{\kappa^2}{2{\dot\varphi}_0 {\bar\sigma}}
++\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
-\biggl[\frac{\partial W^'}{\partial \varpi} \biggr]
-+ \biggl[\frac{\kappa^2 (\varpi {\dot\varphi}^' )}{{\bar\sigma}^2} \biggr]
 </math>
    </td>
 </tr>
+<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
 <tr>
    <td align="right">
-<math>~\Rightarrow ~~~~(\varpi {\dot\varphi}^') (\kappa^2 - {\bar\sigma}^2 ) </math>
+<math>~
+~{\dot{z}}^'
+</math>
    </td>
    <td align="center">
@@ Line 470: / Line 512: @@
    </td>
    <td align="left">
-<math>~\frac{ m{\bar\sigma}^2W^'}{\varpi}
+<math>~
-+ \frac{\kappa^2{\bar\sigma}}{2{\dot\varphi}_0 }
+i~\frac{\partial W^'}{\partial z} \, .
-\biggl[\frac{\partial W^'}{\partial \varpi} \biggr]
 </math>
    </td>
 </tr>
 </table>
-</div>
+where,
+<table border="0" cellpadding="5" align="center">
+<tr>
+  <td align="right">
+<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
+  </td>
+  <td align="center">
+&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
+  </td>
+  <td align="left">
+<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) </math>
+  </td>
+</tr>
+</table>
+</td></tr>
+</table>
-From our more detailed, [[User:Tohline/Cylindrical_3D#Eulerian_Formulation|accompanying discussion]] we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.
+These three velocity-component expressions  match, respectively, equations (3.14), (3.15), and (3.16).
 =See Also=

Difference between revisions of "User:Tohline/Apps/PapaloizouPringle84"

Revision as of 20:09, 13 March 2016

Contents

Nonaxisymmetric Instability in Papaloizou-Pringle Tori

Linearized Principal Governing Equations in Cylindrical Coordinates

Formulation of Eigenvalue Problem

See Also

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