Difference between revisions of "User:Tohline/Appendix/Ramblings/PPTori"

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(→‎Blaes85: Finished determining amplitude and phase functions)
Line 155: Line 155:
</table>
</table>
</div>
</div>
Now, keeping in mind that, for the time being, we are only interested in examining the shape of the unstable eigenvector in the ''equatorial plane'' of the torus, we can set,
<div align="center">
<math>~\cos\theta ~~ \rightarrow ~~ \pm 1 \, .</math>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\beta^4 m^4}\biggl|\frac{\delta W}{W_0} \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[2\eta^2  - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}\biggr]^2
+ 16\biggl[\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta \biggr]^2 </math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[2^3(n+1)^2\eta^2  - 3(n+1)\eta^2 - (4n+1) \biggr]^2
+ \frac{2^3 \cdot 3\eta^2}{(n+1)}  </math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  \biggr]^2
+ \frac{2^3 \cdot 3\eta^2}{(n+1)}  </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \biggl[\frac{2(n+1)}{\beta m} \biggr]^4 \biggl|\frac{\delta W}{W_0} \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  \biggr]^2
+ 2^7 \cdot 3(n+1)^3\eta^2 \, .  </math>
  </td>
</tr>
</table>
</div>
Also,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 161: Line 225:
   <td align="center" bgcolor="blue">&nbsp;</td>
   <td align="center" bgcolor="blue">&nbsp;</td>
   <td align="right">
   <td align="right">
<math>~f_\ln(\varpi)</math>
<math>~m\phi_m</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 167: Line 231:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) \biggr]</math>
<math>~\tan^{-1}\biggl\
\frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta}
\biggr\}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp;  
&nbsp; &nbsp; &nbsp; &nbsp; over &nbsp; &nbsp; &nbsp; &nbsp;  
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
inner region of the torus;
   </td>
   </td>
</tr>
</tr>
<tr><td colspan="6" align="center">and</td></tr>
<tr><td colspan="6" align="center">while</td></tr>
<tr>
<tr>
   <td align="center" bgcolor="green">&nbsp;</td>
   <td align="center" bgcolor="green">&nbsp;</td>
   <td align="right">
   <td align="right">
<math>~f_\ln(\varpi)</math>
<math>~m\phi_m</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 186: Line 252:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr) \biggr]</math>
<math>~\tan^{-1}\biggl\
\frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta}
\biggr\} - k\pi</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp;  
&nbsp; &nbsp; &nbsp; &nbsp; over &nbsp; &nbsp; &nbsp; &nbsp;  
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
outer region of torus.
   </td>
   </td>
</tr>
</tr>

Revision as of 23:59, 18 February 2016

Stability Analyses of PP Tori

Whitworth's (1981) Isothermal Free-Energy Surface
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As has been summarized in an accompanying chapter — also see our related detailed notes — we have been trying to understand why unstable nonaxisymmetric eigenvectors have the shapes that they do in rotating toroidal configurations. For any azimuthal mode, <math>~m</math>, we are referring both to the radial dependence of the distortion amplitude, <math>~f_m(\varpi)</math>, and the radial dependence of the phase function, <math>~\phi_m(\varpi)</math> — the latter is what the Imamura and Hadley collaboration refer to as a "constant phase locus." Some old videos showing the development over time of various self-gravitating "constant phase loci" can be found here; these videos supplement the published work of Woodward, Tohline & Hachisu (1994).

Here, we focus specifically on instabilities that arise in so-called (non-self-gravitating) Papaloizou-Pringle tori and will draw heavily from two publications: (1) Papaloizou & Pringle (1987), MNRAS, 225, 267The dynamical stability of differentially rotating discs.   III. — hereafter, PPIII — and (2) Blaes (1985), MNRAS, 216, 553Oscillations of slender tori.

PP III

Figure 2 extracted without modification from p. 274 of J. C. B. Papaloizou & J. E. Pringle (1987)

"The Dynamical Stability of Differentially Rotating Discs.   III"

MNRAS, vol. 225, pp. 267-283 © The Royal Astronomical Society

Figure 2 from PP III


Blaes85

His Notation

Blaes (1985) adopts a polytropic equation of state,

<math>~\frac{\rho}{\rho_c} = \Theta_H^n \, ,</math>

which gives rise to (slim tori) equilibrium structures for which (see his equation 1.3),

<math>~\Theta_H</math>

<math>~=</math>

<math>~1 - \frac{1}{\beta^2}\biggl[x^2 + x^3(3\cos\theta - \cos^3\theta) + \mathcal{O}(x^4) \biggr] \, ,</math>

where, the (constant) model parameter,

<math>\beta \equiv \frac{(2n)^{1/2}}{\mathcal{M}_0} \, ,</math>

and <math>~\mathcal{M}_0</math> is the Mach number of the rotational velocity at the torus center. Blaes then adopts a related parameter that is constant on isobaric surfaces, namely,

<math>\eta^2 \equiv 1 - \Theta_H \, ,</math>

which is unity at the surface of the torus and which goes to zero at the cross-sectional center of the torus. Notice that <math>~\eta</math> tracks the "radial" coordinate that measures the distance from the center of the torus; in particular, keeping only the leading-order term in <math>~x</math>, there is a simple linear relationship between <math>~\eta</math> and <math>~x</math>, namely,

<math>~\eta</math>

<math>~=</math>

<math>~[1 - \Theta_H]^{1/2} \approx \frac{x}{\beta} \, .</math>


Analytically Prescribed Eigenvector

From my initial focused reading of the analysis presented by Blaes (1985), I conclude that, in the infinitely slender torus case, unstable modes in PP tori exhibit eigenvectors of the form,

<math>~\frac{\delta W}{W_0} \equiv \biggl[ \frac{W(\eta,\theta)}{C} - 1 \biggr]e^{im\Omega_p t}e^{-y_2 (\Omega_0 t)} </math>

<math>~=</math>

<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi_m(\varpi) + k\theta]} \biggr\} \, ,</math>

where we have written the perturbation amplitude in a manner that conforms with the notation that we have used in Figure 1 of a related, but more general discussion. As is summarized in §1.3 of Blaes (1985), for "thick" (but, actually, still quite thin) tori, "exactly one exponentially growing mode exists for each value of the azimuthal wavenumber <math>~m</math>," and its complex amplitude takes the following form (see his equation 1.10):

<math>~f_m(\eta,\theta)</math>

<math>~=</math>

<math> ~\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] + \mathcal{O}(\beta^3) \, . </math>

We should therefore find that the amplitude (modulus) of the enthalpy perturbation is,

<math>~\biggl|\frac{\delta W}{W_0} \biggr|</math>

<math>~=</math>

<math>~\sqrt{[\mathrm{Re}(f_m)]^2+ [\mathrm{Im}(f_m)]^2} \, ;</math>

and the associated phase function should be,

<math>~m\phi_m + k\theta</math>

<math>~=</math>

<math>~\tan^{-1} \biggl\{ \frac{-\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr\}</math>

Now, keeping in mind that, for the time being, we are only interested in examining the shape of the unstable eigenvector in the equatorial plane of the torus, we can set,

<math>~\cos\theta ~~ \rightarrow ~~ \pm 1 \, .</math>

Hence, we have,

<math>~\frac{1}{\beta^4 m^4}\biggl|\frac{\delta W}{W_0} \biggr|^2</math>

<math>~=</math>

<math>~\biggl[2\eta^2 - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}\biggr]^2 + 16\biggl[\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta \biggr]^2 </math>

 

<math>~=</math>

<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[2^3(n+1)^2\eta^2 - 3(n+1)\eta^2 - (4n+1) \biggr]^2 + \frac{2^3 \cdot 3\eta^2}{(n+1)} </math>

 

<math>~=</math>

<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 \biggr]^2 + \frac{2^3 \cdot 3\eta^2}{(n+1)} </math>

<math>~\Rightarrow ~~~~ \biggl[\frac{2(n+1)}{\beta m} \biggr]^4 \biggl|\frac{\delta W}{W_0} \biggr|^2</math>

<math>~=</math>

<math>~\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 \biggr]^2 + 2^7 \cdot 3(n+1)^3\eta^2 \, . </math>

Also,

 

<math>~m\phi_m</math>

<math>~=</math>

<math>~\tan^{-1}\biggl\{ \frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta} \biggr\}</math>

        over        

inner region of the torus;

while
 

<math>~m\phi_m</math>

<math>~=</math>

<math>~\tan^{-1}\biggl\{ \frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2 }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta} \biggr\} - k\pi</math>

        over        

outer region of torus.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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