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.

Formulation of Eigenvalue Problem

From our more detailed, accompanying discussion we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.

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