Difference between revisions of "User:Tohline/Appendix/Ramblings/Azimuthal Distortions"

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(Begin showing empirically derived and analytically defined eigenvector)
(→‎Empirical Construction of Eigenvector: Finished showing how arc-hyperbolic tangent can be rewritten as natural log)
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'''<font color="maroon">PRACTICAL IMPLEMENTATION:</FONT>'''  &nbsp; At the two limits, <math>~\varpi = r_-</math> and <math>~\varpi = r_+</math>, the function, <math>~f(\varpi) \rightarrow +\infty</math>; while, at the limit, <math>~\varpi = r_\mathrm{mid}</math>, the function, <math>~f(\varpi) \rightarrow -\infty</math>.  In practice we stay ''half of a radial zone'' away from these three limiting radial boundaries, so that the maximum and minimum values of <math>~f(\varpi)</math> are finite; then we strategically employ the finite values of the function at these near-boundary limits to rescale the function such that, in the plot shown below, it lies between zero (minimum amplitude) and unity (maximum amplitude).
'''<font color="maroon">PRACTICAL IMPLEMENTATION:</FONT>'''  &nbsp; At the two limits, <math>~\varpi = r_-</math> and <math>~\varpi = r_+</math>, the function, <math>~f(\varpi) \rightarrow +\infty</math>; while, at the limit, <math>~\varpi = r_\mathrm{mid}</math>, the function, <math>~f(\varpi) \rightarrow -\infty</math>.  In practice we stay ''half of a radial zone'' away from these three limiting radial boundaries, so that the maximum and minimum values of <math>~f(\varpi)</math> are finite; then we strategically employ the finite values of the function at these near-boundary limits to rescale the function such that, in the plot shown below, it lies between zero (minimum amplitude) and unity (maximum amplitude).
</td></tr>
</td></tr>
</table>
</div>
Now, the following general relation holds:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tanh^{-1}x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \ln\biggl( \frac{1+x}{1-x} \biggr) </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>x^2 < 1 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, for the innermost region of the toroidal configuration, we can set,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1+x}{1-x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl[2 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]
\biggl[2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]^{-1}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~[(r_\mathrm{mid}-r_-) - ( \varpi - r_-)]  [(\varpi - r_-)]^{-1}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \, .
</math>
  </td>
</tr>
</table>
</div>
Therefore we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr)
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
  </td>
</tr>
</table>
</div>
and, similarly, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_+} \biggr)
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
</table>
</div>
</div>

Revision as of 03:37, 5 January 2016

Analyzing Azimuthal Distortions

Whitworth's (1981) Isothermal Free-Energy Surface
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Adopted Notation

We will adopt the notation of J. E. Tohline & I. Hachisu (1988, ApJ, 361, 394). Specifically, drawing on their equation (2) but ignoring variations in the vertical coordinate, the mass density is given by the expression,

<math>~\rho</math>

<math>~=</math>

<math>~\rho_0 \biggl[ 1 + f(\varpi)e^{-i(\omega t - m\phi)} \biggr] \, ,</math>

where it is understood that <math>~\rho_0</math>, which defines the structure of the initial axisymmetric equilibrium configuration, is generally a function of the cylindrical radial coordinate, <math>~\varpi</math>.

Using the subscript, <math>~m</math>, to identify the time-invariant coefficients and functions that characterize the intrinsic eigenvector of each azimuthal eigen-mode, and acknowledging that the associated eigenfrequency will in general be imaginary, that is,

<math>~\omega_m</math>

<math>~=</math>

<math>~\omega_R + i\omega_I \, ,</math>

we expect each unstable mode to display the following behavior:

<math>~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]</math>

<math>~=</math>

<math>~f_m(\varpi)e^{-i[\omega_R t + i \omega_I t - m\phi_m(\varpi)]} </math>

 

<math>~=</math>

<math>~\biggl\{ f_m(\varpi)e^{-im\phi_m(\varpi)}\biggr\} e^{-i\omega_R t } \cdot e^{\omega_I t} </math>

 

<math>~=</math>

<math>~\biggl\{ f_m(\varpi)e^{-i[\omega_R t + m\phi_m(\varpi)]} \biggr\} e^{\omega_I t} \, .</math>

Adopting Kojima's (1986) notation, that is, defining,

<math>~y_1 \equiv \frac{\omega_R}{\Omega_0} - m</math>

        and        

<math>~y_2 \equiv \frac{\omega_I}{\Omega_0} \, ,</math>

the eigenvector's behavior can furthermore be described by the expression,

<math>~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]</math>

<math>~=</math>

<math>~\biggl\{ f_m(\varpi)e^{-i[(y_1+m) (\Omega_0 t) + m\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} </math>

 

<math>~=</math>

<math>~\biggl\{ f_m(\varpi)e^{-im[(y_1/m+1) (\Omega_0 t) + \phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, .</math>

Note that, as viewed from a frame of reference that is rotating with the mode pattern frequency,

<math>\Omega_p \equiv \frac{\omega_R}{m} = \Omega_0\biggl(\frac{y_1}{m}+1\biggr) \, ,</math>

we should find an eigenvector of the form,

<math>~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]_\mathrm{rot} \equiv \biggl[ \frac{\rho}{\rho_0} - 1 \biggr]e^{im\Omega_p t}</math>

<math>~=</math>

<math>~\biggl\{ f_m(\varpi)e^{-im[\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, ,</math>

whose relative amplitude — with a radial structure as specified inside the curly braces — is undergoing a uniform exponential growth but is otherwise unchanging.

Empirical Construction of Eigenvector

Four panels from figure 2 extracted from p. 252 of J. W. Woodward, J. E. Tohline & I. Hachisu (1994)

"The Stability of Thick, Self-gravitating Disks in Protostellar Systems"

ApJ, vol. 420, pp. 247-267 © American Astronomical Society

Rearranged Figure 2 from Woodward, Tohline, and Hachisu (1994)

As displayed here, the layout of figure panels (a, b, c, d) has been modified from the original publication layout; otherwise, each panel is unmodified.


First, specify a "midway" radial location, <math>~r_- < r_\mathrm{mid} < r_+ \, ,</math> at which the density fluctuation is smallest. Then define a function of the form,

<math>~f(\varpi)</math>

<math>~=</math>

<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) \biggr]</math>

        for        

<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>

and

<math>~f(\varpi)</math>

<math>~=</math>

<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr) \biggr]</math>

        for        

<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>

As shown by the following figure montage, this <math>~f(\varpi)</math> function very closely resembles the one generated by Imamura via a linear stability analysis.

PRACTICAL IMPLEMENTATION:   At the two limits, <math>~\varpi = r_-</math> and <math>~\varpi = r_+</math>, the function, <math>~f(\varpi) \rightarrow +\infty</math>; while, at the limit, <math>~\varpi = r_\mathrm{mid}</math>, the function, <math>~f(\varpi) \rightarrow -\infty</math>. In practice we stay half of a radial zone away from these three limiting radial boundaries, so that the maximum and minimum values of <math>~f(\varpi)</math> are finite; then we strategically employ the finite values of the function at these near-boundary limits to rescale the function such that, in the plot shown below, it lies between zero (minimum amplitude) and unity (maximum amplitude).


Now, the following general relation holds:

<math>~\tanh^{-1}x</math>

<math>~=</math>

<math>~\frac{1}{2} \ln\biggl( \frac{1+x}{1-x} \biggr) </math>

        for        

<math>x^2 < 1 \, .</math>

Hence, for the innermost region of the toroidal configuration, we can set,

<math>~x</math>

<math>~=</math>

<math>~1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) </math>

<math>~\Rightarrow ~~~~ \frac{1+x}{1-x}</math>

<math>~=</math>

<math> ~\biggl[2 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr] \biggl[2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]^{-1} </math>

 

<math>~=</math>

<math> ~[(r_\mathrm{mid}-r_-) - ( \varpi - r_-)] [(\varpi - r_-)]^{-1} </math>

 

<math>~=</math>

<math> ~\frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \, . </math>

Therefore we can write,

<math>~f(\varpi)</math>

<math>~=</math>

<math>~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr) </math>

        for        

<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>

and, similarly, we find,

<math>~f(\varpi)</math>

<math>~=</math>

<math>~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_+} \biggr) </math>

        for        

<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>

Comparison with Immamura

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