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

PP84

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 from PP84

<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, the square of the epicyclic frequency, <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.

Let's make the substitution,

<math>Q_{JT} \equiv (\sigma + m{\dot\varphi}_0) W^' \, ,</math>

in which case,

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

Then we can rewrite the radial component of the perturbed velocity as,

<math>~-i~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>	<math>~=</math>	<math>~\biggl[ (\sigma + m{\dot\varphi}_0)\frac{\partial Q_{JT} }{\partial \varpi} - m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr] +\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) m Q_{JT} </math>
	<math>~=</math>	<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial Q_{JT} }{\partial \varpi} +m Q_{JT} \biggl[ \biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) -\biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr] </math>
	<math>~=</math>	<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial Q_{JT} }{\partial \varpi} +m Q_{JT} \biggl\{ \frac{1}{\varpi^2} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] -\biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr\} </math>
	<math>~=</math>	<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial Q_{JT} }{\partial \varpi} +m Q_{JT} \biggl[ \frac{2\dot\varphi_0}{\varpi} \biggr] </math>

GGN86

In §2.2 of their paper, P. Goldreich, J. Goodman, and R. Narayan (1986, MNRAS, 221, 339) — hereafter, GGN86 — also present expressions for the three components of the perturbed velocity.

Nod to Oort Constants and Simple Rotation Profiles

Acknowledging the galactic dynamics community's familiarity with the so-called Oort constants, and in anticipation of our review of the GGN86 derivation that follows, we define the following two functions:

<math>~A</math>	<math>~\equiv</math>	<math>~- \frac{1}{2}\biggl[ {\dot\varphi}_0 - \frac{\partial}{\partial \varpi}\biggl( \varpi {\dot\varphi}_0 \biggr) \biggr] \, ,</math>
<math>~B</math>	<math>~\equiv</math>	<math>~\frac{1}{2}\biggl[ {\dot\varphi}_0 + \frac{\partial}{\partial \varpi}\biggl( \varpi {\dot\varphi}_0 \biggr) \biggr] \, .</math>

Given these definitions, we note that,

<math>~B - A </math>

<math>~=</math>

<math>~{\dot\varphi}_0 \, ;</math>

and, given the definition of the square of the epicyclic frequency, above, we can write,

<math>~\kappa^2</math>

<math>~=</math>

<math>~4{\dot\varphi}_0 B = 4B (B - A) \, .</math>

In line with our own discussion of simple rotation profiles, GGN86 adopt a generalized power-law rotation profile of the form (see their equation 2.1),

<math>~{\dot\varphi}_0(\varpi)</math>

<math>~=</math>

<math>~ \Omega_0 \biggl( \frac{\varpi}{\varpi_0} \biggr)^{-q} \, ,</math>

in which case we also have,

<math>~\frac{\partial}{\partial \varpi} \biggl( \varpi {\dot\varphi}_0 \biggr)</math>

<math>~=</math>

<math>~ \frac{\partial}{\partial \varpi} \biggl[ \Omega_0 \varpi_0^{q} \varpi^{1-q}\biggr] = (1-q){\dot\varphi}_0 \, .</math>

Given this particular adopted profile, it is therefore clear that,

<math>~A_\mathrm{GGN}</math>	<math>~\equiv</math>	<math>~- \frac{1}{2}\biggl[ {\dot\varphi}_0 - (1-q) {\dot\varphi}_0\biggr] = - \frac{q}{2} {\dot\varphi}_0 \, ;</math>
<math>~B_\mathrm{GGN}</math>	<math>~\equiv</math>	<math>~ \frac{1}{2}\biggl[ {\dot\varphi}_0 + (1-q) {\dot\varphi}_0\biggr] = \frac{1}{2} (2-q) {\dot\varphi}_0 \, ;</math>
<math>~\kappa^2_\mathrm{GGN}</math>	<math>~=</math>	<math>~4{\dot\varphi}_0 \biggl[ \frac{1}{2} (2-q) {\dot\varphi}_0 \biggr] = 2(2-q){\dot\varphi}_0^2 \, .</math>

These three expressions are in line with GGN86 equations (2.4), (2.6), and (2.24), respectively.

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Velocity Components

Specifically, from their equations (2.21) - 2.25) we find,

Perturbed Velocity Components from §2.2 of GGN86

<math>~x</math> Component <math>~ u ( \kappa^2 - \sigma^2_\mathrm{GGN})</math> <math>~=</math> <math>~i \biggl[ \sigma_\mathrm{GGN}~\frac{\partial Q}{\partial x} - 2\Omega_0 k Q \biggr] \, , </math> <math>~y</math> Component <math>~ v ( \kappa^2 - \sigma^2_\mathrm{GGN})</math> <math>~=</math> <math>~2B~\frac{\partial Q}{\partial x} - \sigma_\mathrm{GGN} k Q \, , </math> <math>~z</math> Component <math>~ ~w </math> <math>~=</math> <math>~- i \biggl(\frac{1}{\sigma_\mathrm{GGN}}\biggr) \frac{\partial Q}{\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>

Likely transformations:

• <math>~\sigma_\mathrm{GGN} \equiv \biggl( \omega - \frac{2Amx}{\varpi_0} \biggr) ~~~ \leftrightarrow ~~~ \bar\sigma \equiv (\sigma_\mathrm{Blaes} + m{\dot\varphi}_0 )</math>
• <math>\frac{Q}{\sigma_\mathrm{GGN}} ~~~ \leftrightarrow ~~~ - W^'</math>

Let's take the equations one at a time:

<math>~z</math> Component
<math>~ ~w </math>	<math>~=</math>	<math>~- i \biggl(\frac{1}{\sigma_\mathrm{GGN}}\biggr) \frac{\partial Q}{\partial z} \, . </math>
	<math>~=</math>	<math>~+ i \frac{\partial W^'}{\partial z} \, , </math>

which matches the Blaes85 equation.

<math>~y</math> Component
<math>~ v ( \sigma^2_\mathrm{GGN} - \kappa^2)</math>	<math>~=</math>	<math>~\sigma_\mathrm{GGN} k Q - 2B~\frac{\partial Q}{\partial x} </math>
	<math>~=</math>	<math>~\sigma_\mathrm{GGN} k Q - 2B~\frac{\partial Q}{\partial x} </math>

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