User:Tohline/Appendix/Ramblings/To Hadley and Imamura - Revision history

Tohline: Add introductory "warning" paragraph

2016-05-24T20:54:35Z

Add introductory "warning" paragraph

← Older revision		Revision as of 20:54, 24 May 2016
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	=Summary for Hadley & Imamura=		=Summary for Hadley & Imamura=

			<font color="red"><b>[Comment by J. E. Tohline on 24 May 2016]</b></font>   This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "[[User:Tohline/Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori\|Characteristics of Unstable Eigenvectors in Self-Gravitating Tori]]," that contains a much more trustworthy analysis of this very interesting problem.

			{{LSU_WorkInProgress}}

	This MediaWiki-based document is especially provided for Jimmy Imamura and Kathryn Hadley. It summarizes the contents of a much longer [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Analyzing_Azimuthal_Distortions\|set of technical notes that discusses the analysis of nonaxisymmetric distortions in rotating, self-gravitating fluids]]. The punchline is provided by the [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Put_It_Together\|animation sequence shown in Figure 2, below]].		This MediaWiki-based document is especially provided for Jimmy Imamura and Kathryn Hadley. It summarizes the contents of a much longer [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Analyzing_Azimuthal_Distortions\|set of technical notes that discusses the analysis of nonaxisymmetric distortions in rotating, self-gravitating fluids]]. The punchline is provided by the [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Put_It_Together\|animation sequence shown in Figure 2, below]].

Tohline: /* Comparisons */ work a bit on n=3/2 case

2016-03-08T22:33:19Z

Comparisons: work a bit on n=3/2 case

Show changes

Tohline: /* Various Thoughts */ Add a couple of more concluding "thoughts"

2016-03-08T21:21:22Z

Various Thoughts: Add a couple of more concluding "thoughts"

← Older revision		Revision as of 21:21, 8 March 2016
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	====Various Thoughts====		====Various Thoughts====
	In the Blaes85 model, the "constant phase locus" swings through a total angle of,
			<ol>
			<li>In the Blaes85 model, the "constant phase locus" swings through a total angle of,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	For example, in the model displayed, above, <math>~[\mathcal{A}/\mathcal{B}]_{\eta=1} = 0.4583</math>. Hence, <math>~2\Delta = 4.001</math> radians.		For example, in the model displayed, above, <math>~[\mathcal{A}/\mathcal{B}]_{\eta=1} = 0.4583</math>. Hence, <math>~2\Delta = 4.001</math> radians.</li>

			<li>
			We need to resolve the apparent discrepancy between the value of the leading constant that appears in the GGN86 eigenfunction versus one found in the Blaes85 eigenfunction (when evaluated for <math>~n=0</math>. Graphically, the Blaes85 amplitude function appears to make more sense, but a physically based explanation needs to be identified.
			</li>

			<li>
			The amplitude and phase functions obtained from the Blaes85 work appear to match — qualitatively, if not quantitatively — the amplitude and phase functions published by the Imamura & Hadley collaboration if we subtract the leading "unity" constant from the Blaes85 expression for <math>~W/W_0</math>. On the other hand, when Blaes refers to the modulus of the amplitude, he includes this leading "unity" constant in evaluating the "real" component of his expression. I do not yet fully understand why both ways of viewing the eigenfunction's amplitude — that is, both with and without including the unity term — can be physically relevant.
			</li>
			</ol>

	=See Also=		=See Also=

Tohline: /* Comparisons */ Finished n=0 comparison with GGN86

2016-03-08T20:29:33Z

Comparisons: Finished n=0 comparison with GGN86

← Older revision		Revision as of 20:29, 8 March 2016
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{A}(\eta,\theta)</math>		<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~1 +~~ \frac{ (\beta m)^2}{2^2}\biggl[a(\eta,\theta)\biggr]		<math>~\frac{ (\beta m)^2}{2^2}\biggl[a_0(\eta,\theta)\biggr]
	\, ,		\, ,
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{B}(\eta,\theta)</math>		<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{(\beta m)^2}{2^2}\biggl[b(\eta,\theta)\biggr] \, ,		<math>~\frac{(\beta m)^2}{2^2}\biggl[b_0(\eta,\theta)\biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~a(\eta,\theta)</math>		<math>~a_0(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~\biggl\{
	~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1)		~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=0}
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~b(\eta,\theta)</math>		<math>~b_0(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ~~\biggr~~]^{1/2}		<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=0}
	</math>		</math>
	</td>		</td>
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	</div>		</div>

	From the Blaes85 analysis, then, we ~~should~~ conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,		From the Blaes85 analysis, then, we conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl( \frac{~~\delta~~ W}{W_0}\biggr)_{n=0}</math>		<math>~\biggl[ \frac{W}{W_0}\biggr]_{n=0}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</div>		</div>

	where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29\|our mapping of the GGN86 terminology to the Blaes85 terminology]].		where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29\|our mapping of the GGN86 terminology to the Blaes85 terminology]]. The two derived expressions match in every detail, except one: The constant term, <math>~\tfrac{1}{4}(\beta m)^2</math>, that is subtracted from unity on the right-hand side of the Blaes85 function is missing from the function derived by GGN86. Hopefully, additional study of this problem will rectify this apparent difference between the two published analyses.


			<!-- OMIT
	In addition, the "constant phase locus" should be identified by the function,		In addition, the "constant phase locus" should be identified by the function,

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	</table>		</table>
	</div>		</div>
			END OMIT -->

Tohline: /* Comparisons */ Finished n = 3 evaluation

2016-03-08T20:15:16Z

Comparisons: Finished n = 3 evaluation

← Older revision		Revision as of 20:15, 8 March 2016
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{A}(\eta,\theta)</math>		<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~1 +~~ \frac{ (\beta m)^2}{2^6}\biggl[a(\eta,\theta)\biggr]		<math>~\frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr]
	\, ,		\, ,
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{B}(\eta,\theta)</math>		<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{(\beta m)^2}{2^6}\biggl[b(\eta,\theta)\biggr] \, ,		<math>~\frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~a(\eta,\theta)</math>		<math>~a_3(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>~\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~\biggl\{
	~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1)		~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3}
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~b(\eta,\theta)</math>		<math>~b_3(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[2^7\cdot 3(n+1)^3		<math>~\biggl\{ [2^7\cdot 3(n+1)^3
	\eta^2\cos^2\theta ~~\biggr~~]^{1/2}		\eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\biggl~~[2^{13}\cdot 3\eta^2\cos^2\theta ~~\biggr~~]^{1/2}\, .		<math>~[2^{13}\cdot 3\eta^2\cos^2\theta ]^{1/2}\, .
	</math>		</math>
	</td>		</td>
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	</div>		</div>

			<!-- OMIT
	Hence, the "constant phase locus" should be identified by the function,		Hence, the "constant phase locus" should be identified by the function,

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	</div>		</div>

	and the square of the amplitude is,		END OMIT -->

			Now, rather than examining the structural behavior of the amplitude and phase of the function, <math>~[W/W_0 -1]</math>, as we have done, above, Blaes evaluated the amplitude (only) of the function,
			<div align="center">
			<math>~\frac{W}{W_0} = 1 + \mathcal{A}_\mathrm{Blaes}(\eta,\theta) + i \mathcal{B}_\mathrm{Blaes}(\eta,\theta) \, .</math>
			</div>

			For the specific case of <math>~n=3</math>, the square of the amplitude of ''this'' function is,

	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl\|\frac{~~\delta~~ W}{W_0} \biggr\|^2</math>		<math>~\biggl\|\frac{W}{W_0} \biggr\|_{n=3}^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^6}\biggl[a(\eta,\theta)\biggr] \biggr\}^2 +		<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr] \biggr\}^2 +
	\biggl\{ \frac{(\beta m)^2}{2^6}\biggl[b(\eta,\theta)\biggr] \biggr\}^2		\biggl\{ \frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \biggr\}^2
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1 + \frac{ (\beta m)^2a}{2^5}		<math>~1 + \frac{ (\beta m)^2a_3}{2^5}
	+ \frac{ (\beta m)^4 }{2^{12}} (a^2 + b^2)		+ \frac{ (\beta m)^4 }{2^{12}} (a_3^2 + b_3^2)
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl\|\frac{~~\delta~~ W}{W_0} \biggr\|</math>		<math>~\biggl\|\frac{W}{W_0} \biggr\|_{n=3}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</div>		</div>

	Our expression differs from the one published in Blaes85 in one detail: In Blaes85, the pre-factor of the second term inside the curly braces is, evidently, <math>~(\beta m)^2/2^5</math>, which is a factor of two larger than the corresponding pre-factor found in our expression. In our attempt to re-derive the Blaes85 expression (1.12), this extra factor of two disappears when we take the square-root of both sides to obtain the modulus, rather than the square of the modulus. Hopefully further reflection will resolve this discrepancy between our approximate expression for <math>~\|~~\delta~~ W/W_0\|</math> and the analogous one presented in Blaes85.		Our expression differs from the one published in Blaes85 in one detail: In Blaes85, the pre-factor of the second term inside the curly braces is, evidently, <math>~(\beta m)^2/2^5</math>, which is a factor of two larger than the corresponding pre-factor found in our expression. In our attempt to re-derive the Blaes85 expression (1.12), this extra factor of two disappears when we take the square-root of both sides to obtain the modulus, rather than the square of the modulus. Hopefully further reflection will resolve this discrepancy between our approximate expression for <math>~\|W/W_0\|_{n=3}</math> and the analogous one presented in Blaes85.

	====For n = 0 Configurations====		====For n = 0 Configurations====

Tohline: /* Analytic Analysis of Zero-Mass Papaloizou-Pringle Tori */ Describe movie (Figure 3)

2016-03-08T19:49:07Z

Analytic Analysis of Zero-Mass Papaloizou-Pringle Tori: Describe movie (Figure 3)

← Older revision		Revision as of 19:49, 8 March 2016
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl\{ f_m(\varpi)e^{-im\~~phi_m~~} \biggr\} \, .</math>		<math>~\biggl\{ f_m(\varpi)e^{-im\phi} \biggr\} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\|f_m(\varpi)\|e^{-i[\alpha(\varpi) + \pi/2+ m\~~phi_m~~]} \, . </math>		<math>~\|f_m(\varpi)\|e^{-i[\alpha(\varpi) + \pi/2+ m\phi]} \, . </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~phi_m~~ = \phi_\mathrm{max}(\varpi)</math>		<math>~\phi = \phi_\mathrm{max}(\varpi)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{\~~delta W~~}{W_0} -1</math>		<math>~\frac{W(\eta,\theta)}{W_0} -1</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\~~phi_m~~ + k\theta]} \biggr\} \, ,</math>		<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi + k\theta]} \biggr\} \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{A}(\eta,\theta)</math>		<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathcal{B}(\eta,\theta)</math>		<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</div>		</div>

	We should therefore find that the amplitude (modulus) of the ~~enthalpy~~ perturbation is,		We should therefore find that the amplitude (modulus) of the perturbation is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\biggl\|\frac{~~\delta~~ W}{W_0} \biggr\|</math>		<math>~\biggl\|\frac{W}{W_0} -1\biggr\|</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\sqrt{\mathcal{A}^2+ \mathcal{B}^2} \, ;</math>		<math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2+ \mathcal{B}_\mathrm{Blaes}^2} \, ;</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\tan^{-1}\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr] - k\theta		\tan^{-1}\biggl[ \frac{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr] - k\theta
	\, .</math>		\, .</math>
	</td>		</td>
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	====Movie====		====Movie====

			Figure 3 displays, for various values of the polytropic index, <math>~0.0 \le n\le 5.0</math>, the radial structure of the Blaes85 <math>~m=2</math> eigenfunction for a slim PP-torus having <math>~\beta = 0.18</math>. More specifically, for each frame of the animation, the figure displays as a function of the dimensionless radial coordinate, <math>~x</math>, (left panel) the behavior of
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="2" ~~bordercolor~~="~~red~~">		'''Left:'''    <math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2 + \mathcal{B}_\mathrm{Blaes}^2}</math>
	<tr><td align="center" bgcolor="~~darkblue~~">		</div>
			on a semi-log plot; and (right panel) the "constant phase locus,"
			<div align="center">
			'''Right:'''    <math>~\frac{1}{2} \tan^{-1} \biggl[ \frac{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr] - \theta \, .</math>
			</div>

			<div align="center">
			<table border="1" align="center" cellpadding="2" width="610px">
			<tr>
			<td align="center" colspan="2">
			<font size="+1"><b>Figure 3:</b></font>   The Blaes85 Analytic Eigenfunction
			</td>
			</tr>

			<tr><td align="center" bgcolor="gray" colspan="2">
	[[File:Blaes85Analytic.gif\|600px\|Amp and Phase from Blaes (1985)]]		[[File:Blaes85Analytic.gif\|600px\|Amp and Phase from Blaes (1985)]]
	</td></tr>		</td></tr>
			<tr>
			<td align="left" colspan="2">
			CAPTION:   Radial dependence of the (left) amplitude and (right) "constant phase locus" of the Blaes85 analytic eigenfunction, for various values of the polytropic Index, <math>~n</math>, as indicated in the bottom-right corner of the left panel.
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Blaes (1985) */ Get rid of leading "unity" term

2016-03-07T19:01:31Z

Blaes (1985): Get rid of leading "unity" term

← Older revision		Revision as of 19:01, 7 March 2016
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{\delta W}{W_0} </math>		<math>~\frac{\delta W}{W_0} -1</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>		<math>
	~~~1 +~~ \beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}		~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}
	\pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr]		\pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr]
	+ \mathcal{O}(\beta^3) \, .		+ \mathcal{O}(\beta^3) \, .
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~1 +~~ (\beta m)^2\biggl[		<math>~(\beta m)^2\biggl[
	2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr]		2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr]
	</math>		</math>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~1 +~~ \frac{ (\beta m)^2}{4(n+1)^2}\biggl\{		<math>~\frac{ (\beta m)^2}{4(n+1)^2}\biggl\{
	~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}		~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}
	\, ;		\, ;

Tohline: /* Blaes (1985) */ Insert animation sequence Blaes85Analytic.gif

2016-03-07T18:09:24Z

Blaes (1985): Insert animation sequence Blaes85Analytic.gif

← Older revision		Revision as of 18:09, 7 March 2016
Line 882:		Line 882:
	* Because (at least for slim PP-tori) the ratio, <math>~\mathcal{A}/\mathcal{B}</math> contains an overall factor that is an odd power of <math>~\cos\theta</math>, the argument of the arctangent function will automatically flip signs as we move from the "inner" region of the torus to the "outer" region of the torus.		* Because (at least for slim PP-tori) the ratio, <math>~\mathcal{A}/\mathcal{B}</math> contains an overall factor that is an odd power of <math>~\cos\theta</math>, the argument of the arctangent function will automatically flip signs as we move from the "inner" region of the torus to the "outer" region of the torus.
	* We now appreciate that a plot of "constant phase locus" is smooth across the mid-point (across the cross-sectional center) of the torus because the definition of <math>~\phi_\mathrm{max}</math> naturally contains a phase shift of <math>~k\theta</math>.		* We now appreciate that a plot of "constant phase locus" is smooth across the mid-point (across the cross-sectional center) of the torus because the definition of <math>~\phi_\mathrm{max}</math> naturally contains a phase shift of <math>~k\theta</math>.

			====Movie====

			<div align="center">
			<table border="1" align="center" cellpadding="2" bordercolor="red">
			<tr><td align="center" bgcolor="darkblue">
			[[File:Blaes85Analytic.gif\|600px\|Amp and Phase from Blaes (1985)]]
			</td></tr>
			</table>
			</div>

	===Comparisons===		===Comparisons===

Tohline: /* Comparisons */ Finished (?) detailing all three values of polytropic index, that is, 3, 1.5, and 0

2016-03-07T01:44:27Z

Comparisons: Finished (?) detailing all three values of polytropic index, that is, 3, 1.5, and 0

← Older revision		Revision as of 01:44, 7 March 2016
Line 1,105:		Line 1,105:

	====For n = 0 Configurations====		====For n = 0 Configurations====

			As we have [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29\|discussed elsewhere]], [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman & Narayan (1986, MNRAS, 221, 339)] — hereafter, GGN86 — also used analytic techniques to analyze the properties of unstable, nonaxisymmetric eigenmodes in Papaloizou-Pringle tori. They restricted their discussion to slim, ''incompressible'' tori, so in order to assess the overlap between the GGN86 and Blaes85 works, we will set <math>~n=0</math> in the general expressions presented in Blaes85.

	<div align="center">		<div align="center">
Line 1,197:		Line 1,199:
	</table>		</table>
	</div>		</div>

			From the Blaes85 analysis, then, we should conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl( \frac{\delta W}{W_0}\biggr)_{n=0}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 + \frac{ (\beta m)^2}{2^2}\biggl[(8\cos^2\theta - 3)\eta^2 - 1\biggr]
			+ i \frac{ (\beta m)^2}{2^2}\biggl[ 2^3 (2\cdot 3)^{1/2} \eta \cos\theta \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[1 - \tfrac{1}{4}(\beta m)^2\biggr] + (\beta m)^2 \biggl[\biggl(2\cos^2\theta - \frac{3}{4}\biggr)\eta^2 \biggr]
			+ 4i (\beta m)^2\biggl[ \biggl(\frac{3}{2}\biggr)^{1/2} \eta \cos\theta \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			This should be compared with equation (5.16) of GGN86, which is "the lowest order [complex] expression for the [perturbed] velocity potential," namely,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\psi </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1+m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr]
			\mp 4i \biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29\|our mapping of the GGN86 terminology to the Blaes85 terminology]].


			In addition, the "constant phase locus" should be identified by the function,

			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 + a(\eta,\theta) }{ b(\eta,\theta) } \biggr\} - k\theta
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 - 1
			+ [8\cos^2\theta - 3]\eta^2 }{ 2^3 (2\cdot 3)^{1/2} \eta \cos\theta } \biggr\} - k\theta \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>

			and the square of the amplitude is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\biggl\|\frac{\delta W}{W_0} \biggr\|^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^2}\biggl[a(\eta,\theta)\biggr] \biggr\}^2 +
			\biggl\{ \frac{(\beta m)^2}{2^2}\biggl[b(\eta,\theta)\biggr] \biggr\}^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 + \frac{ 2(\beta m)^2a}{2^2}
			+ \frac{ (\beta m)^4 }{2^4} (a^2 + b^2)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~1 + \frac{ (\beta m)^2}{2} \biggl\{ [8\cos^2\theta - 3]\eta^2 - 1 \biggr\}
			+ \mathcal{O}[(\beta m)^4] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

Tohline: /* Comparisons */ Explain discrepancy regarding Blaes85 equation (1.12)

2016-03-06T23:40:58Z

Comparisons: Explain discrepancy regarding Blaes85 equation (1.12)

Show changes