User:Tohline/Apps/PapaloizouPringle84

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 1st 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 1st 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.

Rewritten Velocity Components

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) of PP84.

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation