User:Tohline/Apps/Blaes85SlimLimit

Stability of PP Tori in the Slim Torus Limit

Statement of the Eigenvalue Problem

Here, we build on our discussion in an accompanying chapter in which five published analyses of nonaxisymmetric instabilities in Papaloizou-Pringle tori were reviewed: The discovery paper, PP84, and papers by four separate groups that were published within a couple of years of the discovery paper — Papaloizou & Pringle (1985), Blaes (1985), Kojima (1986), and Goldreich, Goodman & Narayan (1986). Following the lead of Blaes (1985; hereafter Blaes85), in particular, we have shown that the relevant eigenvalue problem is defined by the following 2^nd-order PDE,

<math>~0</math>

<math>~=</math>

<math>~ \eta^2 (1-\eta^2)\cdot \frac{\partial^2(\delta W)^{(0)}}{\partial \eta^2} + (1-\eta^2) \cdot \frac{\partial^2(\delta W)^{(0)}}{\partial\theta^2} + \biggl[ \eta (1-\eta^2) -2 n \eta^3 \biggr] \cdot \frac{\partial (\delta W)^{(0)}}{\partial \eta} + 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 (\delta W)^{(0)} \, , </math>

where, <math>~\delta W^{(0)}</math> is the dimensionless enthalpy perturbation. Making the substitution,

<math>~\delta W^{(0)} ~\rightarrow~ V(\eta) \exp (ik\theta) \, ,</math>

this governing equation — now, a one-dimensional, 2^nd-order ODE — becomes,

<math>~0</math>

<math>~=</math>

<math>~ \eta^2 (1-\eta^2)\cdot \frac{d^2V}{d \eta^2} - k^2(1-\eta^2) V + \biggl[ \eta (1-\eta^2) -2 n \eta^3 \biggr] \cdot \frac{d V}{d \eta} + 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 V \, . </math>

Making the additional substitution,

<math>~V ~\rightarrow~ \eta^{|k|} \Upsilon(\eta) \, ,</math>

and appreciating that,

<math>~\frac{dV}{d\eta}</math>	<math>~=</math>	<math>~|k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \, ,</math>
<math>~\frac{d^2V}{d\eta^2}</math>	<math>~=</math>	<math>~ |k|[|k|-1] \eta^{|k|-2}\Upsilon + 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} + \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2}\, ,</math>

the governing ODE becomes,

<math>~ \biggl\{k^2(1-\eta^2) - 2n\eta^2 \biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2\biggr\} \eta^{|k|}\Upsilon </math>	<math>~=</math>	<math>~ \eta^2 (1-\eta^2)\cdot \biggl[ |k|[|k|-1] \eta^{|k|-2}\Upsilon + 2|k|\eta^{|k|-1} \frac{d\Upsilon}{d\eta} + \eta^{|k|} \frac{d^2\Upsilon}{d\eta^2} \biggr] + \biggl[ \eta (1-\eta^2) -2 n \eta^3 \biggr] \cdot \biggl[ |k|\eta^{|k|-1} \Upsilon + \eta^{|k|} \frac{d\Upsilon}{d\eta} \biggr] </math>
	<math>~=</math>	<math>~(1-\eta^2) \biggl[ |k|[|k|-1] \eta^{|k|}\Upsilon + 2|k|\eta^{|k|+1} \frac{d\Upsilon}{d\eta} + \eta^{|k|+2} \frac{d^2\Upsilon}{d\eta^2}\biggr] + \biggl[ (1-\eta^2) -2 n \eta^2 \biggr] \cdot \biggl[ |k|\eta^{|k|} \Upsilon + \eta^{|k|+1} \frac{d\Upsilon}{d\eta} \biggr] </math>
	<math>~=</math>	<math>~\eta^{|k|}(1-\eta^2) \biggl[ k^2 \Upsilon + (2|k|+1)\eta \frac{d\Upsilon}{d\eta} + \eta^{2} \frac{d^2\Upsilon}{d\eta^2} \biggr] - \eta^{|k|}\biggl[ 2 n \eta^2 \biggr] \cdot \biggl[ |k| \Upsilon + \eta \frac{d\Upsilon}{d\eta} \biggr] </math>
<math>~\Rightarrow~~~ - 2n\eta^2 \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math>	<math>~=</math>	<math>~(1-\eta^2) \biggl[ \eta^{2} \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)\eta \frac{d\Upsilon}{d\eta} \biggr] - \biggl[ 2 n \eta^3 \frac{d\Upsilon}{d\eta} \biggr] \, . </math>

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Finally, then, making the independent variable substitution,

<math>~\eta^2 ~\rightarrow ~ y</math>       <math>~\Rightarrow</math>       <math>~dy = 2\eta d\eta</math>      

in which case,

<math>~\frac{d}{d\eta}</math>	<math>~\rightarrow</math>	<math>~2y^{1/2}\frac{d}{dy}</math>
<math>~\frac{d^2}{d\eta^2}</math>	<math>~\rightarrow</math>	<math>~2\frac{d}{dy} + 4y\frac{d^2}{dy^2} \, .</math>

and,

<math>~ - 2ny \biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math>	<math>~=</math>	<math>~ (1-y) y \frac{d^2\Upsilon}{d\eta^2} + (2|k|+1)(1-y)y^{1/2} \frac{d\Upsilon}{d\eta} - 2 n y^{3/2} \frac{d\Upsilon}{d\eta} </math>
	<math>~=</math>	<math>~ 4(1-y)y^2 \frac{d^2\Upsilon}{dy^2} + 2(1-y) y \frac{d\Upsilon}{dy} + 2(2|k|+1)(1-y)y \frac{d\Upsilon}{dy} - 4 n y^{2} \frac{d\Upsilon}{dy} </math>
<math>~\Rightarrow~~~~ - \frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m \biggr)^2 -|k|\biggr] \Upsilon </math>	<math>~=</math>	<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + \frac{1}{2}(1-y) \frac{d\Upsilon}{dy} + \frac{1}{2}(2|k|+1)(1-y)\frac{d\Upsilon}{dy} - n y \frac{d\Upsilon}{dy} </math>
	<math>~=</math>	<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)(1-y)\frac{d\Upsilon}{dy} - n y \frac{d\Upsilon}{dy} </math>
	<math>~=</math>	<math>~ (1-y)y \frac{d^2\Upsilon}{dy^2} + (|k|+1)\frac{d\Upsilon}{dy} -y (|k|+1+n)\frac{d\Upsilon}{dy} \, . </math>

This matches equation (3.9) of Blaes85.

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