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Linearized Equations in Cylindrical Coordinates

Whitworth's (1981) Isothermal Free-Energy Surface
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Eulerian Formulation of Nonlinear Governing Equations

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


\varpi Component of Euler Equation


\frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial \dot\varpi}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] -  \varpi {\dot\varphi}^2   = 
- \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}


\varphi Component of Euler Equation



\frac{\partial (\varpi\dot\varphi)}{\partial t} + \biggl[ \dot\varpi \frac{\partial (\varpi\dot\varphi)}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial (\varpi\dot\varphi)}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial (\varpi\dot\varphi)}{\partial z} \biggr] + \dot\varpi \dot\varphi = 
-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr]


z Component of Euler Equation


\frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] 
+ \biggl[ \dot\varphi \frac{\partial \dot{z}}{\partial\varphi} \biggr] +\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] = 
- \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z}


Equation of Continuity


\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0

These match, for example, equations (3.1) - (3.4) of Papaloizou & Pringle (1984, MNRAS, 208, 721-750), hereafter, PPI.

Linearization

If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:

Linearizing Radial Component of Euler Equation

~\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( { {\dot\varphi}_0 + {\dot\varphi}^'})^2

~=

~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}

~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( {\dot\varphi}_0)^2 -  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')

~=

~
- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi} 
- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0}  \biggr) 
- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}

~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr] 
+ \frac{\partial \Phi^'}{\partial \varpi}

~=

~\biggl\{  \varpi ( {\dot\varphi}_0)^2 
- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi} 
- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}

~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr] + \frac{\partial \Phi^'}{\partial \varpi}

~=

~0 \, .

This last expression has been obtained by recognizing that, in the next-to-last expression: (1) The terms inside the curly braces on the right-hand side collectively provide a statement of equilibrium (in the radial-coordinate direction) in the initial, unperturbed configuration and therefore the terms sum to zero; and (2) the terms inside square brackets on the left-hand side can be rewritten in a more compact form because we have adopted a polytropic equation of state to build the unperturbed initial equilibrium configuration and are examining only adiabatic perturbations with ~\gamma = (n+1)/n, in which case,

~\frac{\nabla P_0}{P_0} = \frac{(n+1)}{n} \cdot \frac{\nabla \rho_0}{\rho_0} \, ,

      and      

~\frac{P^'}{P_0} = \frac{\gamma \rho^'}{\rho_0} \, .


Linearizing Azimuthal Component of Euler Equation

Keeping in mind that the initial equilibrium configuration is axisymmetric — that is, equilibrium parameters exhibit no variation in the azimuthal direction — and, in addition, ~\dot\varphi_0 exhibits no variation in the vertical direction, we have,

~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + ( {\dot\varpi}^') \frac{\partial (\varpi\dot\varphi_0)}{\partial\varpi}  + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
( {\dot\varpi}^') {\dot\varphi_0}

~=

~-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial \varphi} + \frac{\partial \Phi^'}{\partial \varphi} \biggr]

~\Rightarrow ~~~~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]

~=

~-  \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr] 
\, .


Linearizing Vertical Component of Euler Equation

~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}

~=

~
- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial z} - \frac{\partial (\Phi_0+\Phi^')}{\partial z}

 

~=

~
- \frac{1}{\rho_0}\frac{\partial P^'}{\partial z} 
- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial z}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0}  \biggr) 
- \frac{\partial (\Phi_0+\Phi^')}{\partial z}

~\Rightarrow~~~~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial z}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial z}\biggr] 
+ \frac{\partial \Phi^'}{\partial z}

~=

~\biggl\{ 
- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial z} 
- \frac{\partial \Phi_0}{\partial z} \biggr\}

~\Rightarrow~~~~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
+ \biggl[ \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr]  
+ \frac{\partial \Phi^'}{\partial z}

~=

~0 \, ,

where the logic followed in deriving the last expression from the next-to-last one is directly analogous to the logic used, above, in obtaining the final expression for the radial component of the linearized Euler equation.

Linearizing Continuity Equation

~\frac{\partial\rho^'}{\partial t}

~=

~
- \frac{1}{\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}^' + \rho^' \varpi {\dot\varphi}_0 \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr]

~\Rightarrow~~~~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi}

~=

~
- \frac{1}{\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}^' \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .

Summary

Set of Linearized Principal Governing Equations in Cylindrical Coordinates

Continuity Equation

~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi}

~=

~
- \frac{1}{\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}^' \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .

\varpi Component of Euler Equation

~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
( {\dot\varphi}_0 ) \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')

~=

~
- \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr)  - \frac{\partial \Phi^'}{\partial \varpi}

\varphi Component of Euler Equation

~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]

~=

~-  \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr]

~z Component of Euler Equation

~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}

~=

~
- \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) 
- \frac{\partial \Phi^'}{\partial z}

Adiabatic Form of the 1st Law of Thermodynamics

~\frac{P^'}{P_0}

~=

~ \frac{\gamma \rho^'}{\rho_0}

Poisson Equation

~\nabla^2 \Phi^'

~=

~
4\pi G\rho^'

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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