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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 {\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.

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. Here we seek to identify key differences in approach but, ultimately, highlight the degree of agreement between the GGN86 and the PP84 analyses.

Preamble

First, 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}^' ( \kappa^2 - {\bar\sigma}^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} + (2\dot\varphi_0)\frac{m Q_{JT}}{\varpi} </math>
<math>~{\dot\varpi}^' ( \kappa^2 - {\bar\sigma}^2 )</math>	<math>~=</math>	<math>~- i~\bar\sigma \frac{\partial Q_{JT} }{\partial \varpi} - (2i\dot\varphi_0)\frac{m Q_{JT}}{\varpi} \, . </math>

Similarly, we can rewrite the azimuthal component of the perturbed velocity as,

<math>~(\varpi {\dot\varphi}^') ( \kappa^2 - {\bar\sigma}^2) </math>

<math>~ \frac{ m{\bar\sigma}^2 }{\varpi}(\sigma + m{\dot\varphi}_0)^{-1}Q_{JT} + \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }

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

<math>~ \frac{ m{\bar\sigma} }{\varpi} Q_{JT} + \frac{\kappa^2 }{ 2{\dot\varphi}_0 }

\biggl\{ ~\biggl[ \frac{\partial  Q_{JT} }{\partial \varpi}

- \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \frac{m}{\bar\sigma} Q_{JT} \biggr] +\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{m}{\bar\sigma } Q_{JT} \biggr\} </math>

<math>~ \biggl( \frac{\kappa^2 }{ 2{\dot\varphi}_0 } \biggr) \frac{\partial Q_{JT} }{\partial \varpi} + \bar\sigma \frac{mQ_{JT} }{\varpi} \, . </math>

Finally, the vertical component of the perturbed velocity becomes,

<math>~ ~{\dot{z}}^' </math>	<math>~=</math>	<math>~ i~(\sigma + m{\dot\varphi}_0)^{-1} \frac{\partial Q_{JT}}{\partial z} \, . </math>
<math>~\Rightarrow~~~~i~\bar\sigma {\dot{z}}^' </math>	<math>~=</math>	<math>~ -\frac{\partial Q_{JT}}{\partial z} \, . </math>

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 "Oort 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>~{\dot\varphi}_0 \, ;</math>

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

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

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

In line with our own discussion of simple rotation profiles (but note that, in that chapter, the variable we use for the power-law exponent is different from theirs), GGN86 adopt a generalized power-law rotation profile of the form (see their equation 2.1),

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

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Velocity Components

Direct Comparison of Derived Equations

The left panel of the following equation-table presents, once again, our above rewrite of the three components of the perturbed velocity derived in PP84 after we have replaced <math>~\kappa^2</math> (on the right-hand side of the <math>~\varphi</math> component) with its expression in terms of the "Oort function", <math>~B</math>. For comparison, the right panel of the same equation-table shows the analogous perturbed velocity-component expressions derived by GGN86 (see their equations 2.21 - 2.25).

Rewrite of the Components of the Perturbed Velocity from PP84

Perturbed Velocity Components from §2.2 of GGN86

<math>\varpi</math> Component
<math>~{\dot\varpi}^' ( \kappa^2 - {\bar\sigma}^2 )</math>	<math>~=</math>	<math>~- i~\bar\sigma \frac{\partial Q_{JT} }{\partial \varpi} - (2i\dot\varphi_0)\frac{m Q_{JT}}{\varpi} \, , </math>
<math>\varphi</math> Component
<math>~(\varpi {\dot\varphi}^') ( \kappa^2 - {\bar\sigma}^2) </math>	<math>~=</math>	<math>~ 2B \frac{\partial Q_{JT} }{\partial \varpi} + \bar\sigma \frac{mQ_{JT} }{\varpi} \, , </math>
<math>~z</math> Component
<math>~\Rightarrow~~~~i~\bar\sigma {\dot{z}}^' </math>	<math>~=</math>	<math>~ -\frac{\partial Q_{JT}}{\partial z} \, . </math>

<math>~x \equiv (\varpi - \varpi_0)</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 \equiv (\varpi_0 \varphi)</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>~ ~-i~\sigma_\mathrm{GGN}w </math>	<math>~=</math>	<math>~- \frac{\partial Q}{\partial z} \, . </math>

These sets of expressions are identical if we adopt the following three variable mappings,

<math>~Q_{JT}\equiv\bar\sigma W^'</math>	<math>~\leftrightarrow</math>	<math>~Q \, ,</math>
<math>~-~\bar\sigma </math>	<math>~\leftrightarrow</math>	<math>~\sigma_\mathrm{GGN} \, ,</math>
<math>~m/\varpi</math>	<math>~\leftrightarrow</math>	<math>~k \, ,</math>

and recognize that the appropriate association between the variable names that has been used for the three perturbed velocity-components is:

<math>~{\dot\varpi}^'</math>	<math>~\leftrightarrow</math>	<math>~u \, ,</math>
<math>~(\varpi {\dot\varphi}^') </math>	<math>~\leftrightarrow</math>	<math>~v \, ,</math>
<math>~{\dot{z}}^'</math>	<math>~\leftrightarrow</math>	<math>~w \, .</math>

Checking Self-Consistency

<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>~\Rightarrow~~~~ -2A</math>	<math>~=</math>	<math>~- \varpi \frac{\partial {\dot\varphi}_0}{\partial \varpi}</math>
<math>~\Rightarrow~~~~ \frac{\partial {\dot\varphi}_0}{\partial \varpi}</math>	<math>~=</math>	<math>~\frac{2A}{\varpi}</math>

Now, expand the function, <math>~{\dot\varphi}_0(\varpi)</math> in a Taylor series …

<math>~{\dot\varphi}_0(\varpi) </math>	<math>~\approx</math>	<math>~\Omega_0 + (\varpi - \varpi_0)\frac{\partial {\dot\varphi}_0}{\partial\varpi}\biggr\|_{\varpi_0}</math>
	<math>~=</math>	<math>~\Omega_0 + (\varpi - \varpi_0)\frac{2A}{\varpi_0}</math>
<math>~\Rightarrow ~~~~\bar\sigma \equiv (\sigma + m{\dot\varphi}_0)</math>	<math>~\approx</math>	<math>~\sigma + \biggl[m\Omega_0 + \frac{2mA}{\varpi_0}(\varpi - \varpi_0)\biggr]</math>

Now, from equations (2.18) and (2.15) of GGN86, along with their definition of the independent variable, <math>~x</math>, we have,

<math>~- \sigma_\mathrm{GGN}</math>	<math>~=</math>	<math>~- \omega_\mathrm{GGN} + 2Akx</math>
	<math>~=</math>	<math>~- \omega_\mathrm{GGN} + \frac{2mA}{\varpi_0} (\varpi-\varpi_0) \, .</math>

Hence, we can understand the desired mapping, <math>\bar\sigma \leftrightarrow - \sigma_\mathrm{GGN}</math>, if we acknowledge the more fundamental mapping,

<math>~\omega_\mathrm{GGN} ~~ \leftrightarrow ~~ - (\sigma+m\Omega_0) \, .</math>

Adopting Kojima's (1986) <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a separate but closely related chapter, we therefore have,

<math>~y_1 </math>	<math>~=</math>	<math>~\frac{\mathrm{Re}(\omega_\mathrm{GGN})}{\Omega_0} - m </math>
	<math>~=</math>	<math>~\biggl( \frac{1}{\Omega_0} \biggr) \mathrm{Re}\biggl[ - (\sigma+m\Omega_0) \biggr] - m </math>
<math>~y_2 </math>	<math>~=</math>	<math>~\frac{\mathrm{Im}(\omega_\mathrm{GGN})}{\Omega_0} \, ,</math>

Formulation of Eigenvalue Problem

Let's plug the three expressions for the components of the perturbed velocity into the linearized continuity equation and, as well, replace <math>~\rho^'</math> in favor of <math>~W^'</math> via the expression,

<math>~W^' </math>	<math>~\equiv</math>	<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
	<math>~=</math>	<math>~\frac{1}{ (\sigma + m{\dot\varphi}_0)} \biggl(\frac{\gamma P_0 \rho^' }{\rho_0^2}\biggr) \, .</math>

Also multiplying through by <math>~-i</math>, we obtain,

<math>~0 </math>	<math>~=</math>	<math>~\rho^'[(\sigma + m{\dot\varphi}_0)] + \frac{m\rho_0}{\varpi} (\varpi {\dot\varphi}^' ) - \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi (i {\dot\varpi}^' ) \biggr] - \frac{\partial}{\partial z} \biggl[ \rho_0 (i{\dot{z}}^') \biggr] </math>
	<math>~=</math>	<math>~{\bar\sigma}^2 \biggl( \frac{\rho_0^2 W^'}{\gamma P_0 } \biggr) + \frac{m\rho_0}{\varpi ({\bar\sigma}^2 - \kappa^2 )} \biggl\{ - \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] \biggr\} </math>
		<math>~ + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{({\bar\sigma}^2 - \kappa^2 )} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) mW^' \bar\sigma \biggr] \biggr\} + \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} \, . </math>

Multiplying through by <math>~D^2 \equiv ({\bar\sigma}^2 - \kappa^2)^2</math> and reorganizing terms gives,

<math>~0 </math>	<math>~=</math>	<math>~ \frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0} \biggr) \biggr] \biggr\} + \frac{D m\rho_0}{\varpi} \biggl\{ - \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] \biggr\} </math>
		<math>~ + D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} + \biggl( \frac{D^2 {\bar\sigma}^2 \rho_0^2 W^'}{\gamma P_0 } \biggr) </math>
	<math>~=</math>	<math>~ \frac{D^2}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi}{D} \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} +\biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0} \biggr) \biggr] \biggr\} + \rho_0 D \biggl\{ - \frac{m^2W^' }{\varpi^2 } \biggl[{\bar\sigma}^2 + \frac{\kappa^2 \varpi }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \biggr] - \frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 } \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\} </math>
		<math>~ + D^2 \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} + \biggl( \frac{D^2 {\bar\sigma}^2 \rho_0^2 W^'}{\gamma P_0 } \biggr) \, . </math>

As can be confirmed by comparing it to equation (3.18) of PP84 — which, to facilitate comparison, has been extracted and displayed in the following framed image — this expression matches the 2^nd-order, two-dimensional PDE that defines the eigenvalue problem discussed by Papaloizou and Pringle in their seminal 1984 paper.

Equation (3.18) extracted without modification from p. 726 of Papaloizou & Pringle (1984)

"The dynamical stability of differentially rotating discs with constant specific angular momentum"

Papaloizou and Pringle (1984, MNRAS, 208, 721)

After dividing through by <math>~D^2</math> and, again, rearranging terms, we also have,

<math>~ \frac{ {\bar\sigma}^2 \rho_0^2 W^'}{\gamma P_0 } </math>	<math>~=</math>	<math>~- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\} -\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{\frac{\rho_0 \varpi}{D} \biggl( \frac{\kappa^2mW^' \bar\sigma }{2\varpi {\dot\varphi}_0} \biggr) \biggr\} </math>
		<math>~ + \frac{\rho_0}{D} \biggl\{ \frac{m^2 {\bar\sigma}^2 W^' }{\varpi^2 } \biggr\} + \frac{\rho_0}{D} \biggl\{ \frac{m^2W^' }{\varpi^2 } \biggl[\frac{\kappa^2 \varpi }{ 2 {\dot\varphi}_0 } \biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \biggr] \biggr\} + \frac{\rho_0}{D} \biggl\{\frac{m\kappa^2 {\bar\sigma} }{ 2\varpi{\dot\varphi}_0 } \biggl[ ~\frac{\partial W^'}{\partial \varpi} \biggr]\biggr\} </math>
		<math>~ - \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} </math>
	<math>~=</math>	<math>~- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\} + \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\} - \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} </math>
		<math>~ -\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} \biggl( W^' \bar\sigma \biggr) \biggr] + \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{W^' }{\varpi} \cdot \frac{\partial (m{\dot\varphi}_0)}{\partial\varpi} \biggr] + \frac{\kappa^2 m \rho_0}{2{\dot\varphi}_0 D} \biggl[\frac{ {\bar\sigma} }{\varpi} \cdot \frac{\partial W^'}{\partial \varpi} \biggr] </math>
	<math>~=</math>	<math>~- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl\{ \frac{\rho_0 \varpi {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr\} + \biggl\{ \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} \biggr\} - \frac{\partial}{\partial z} \biggl\{ \rho_0 \frac{\partial W^'}{\partial z} \biggr\} </math>
		<math>~ -\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0 \kappa^2m }{2 {\dot\varphi}_0 D} ( W^' \bar\sigma ) \biggr] + \frac{\rho_0 \kappa^2 m}{2 {\dot\varphi}_0 D} \biggl[ \frac{1}{\varpi} \cdot \frac{\partial (W^' \bar\sigma)}{\partial\varpi} \biggr] </math>
	<math>~=</math>	<math>~- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl( \frac{\rho_0 \varpi {\bar\sigma}^2}{D} \cdot \frac{\partial W^'}{\partial \varpi} \biggr) + \frac{\rho_0 m^2 {\bar\sigma}^2 W^' }{\varpi^2 D} - \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) - \frac{m W^' \bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \biggl( \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0 } \biggr)\biggr] \, . </math>

If, inside the last term, we make the substitution,

<math>~h^' \equiv \frac{\varpi \kappa^2 }{2 {\dot\varphi}_0} \, ,</math>

this expression matches equation (2.19) of Papaloizou & Pringle (1985) — which, to facilitate comparison, has been extracted and displayed in the following framed image. Clearly this mathematical definition of the eigenvalue problem discussed by PP85 is fundamentally the same as the one introduced and discussed in PP84.

Equation (2.19) extracted without modification from p. 803 of Papaloizou & Pringle (1985)

"The dynamical stability of differentially rotating discs. II"

Papaloizou and Pringle (1985, MNRAS, 213, 799)

Equation (1.6) extracted without modification from p. 555 of Blaes (1985)

"Oscillations of Slender Tori"

Equation (2.27) extracted without modification from p. 343 of Goldreich, Goodman & Narayan (1986)

"The stability of accretion tori. I - Long-wavelength modes of slender tori"

Equations (12) & (13) extracted without modification from p. 254 of Kojima (1986)

"The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum"

Kojima (1986, Progress of Theoretical Physics, 75, 251)

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