Analyzing Azimuthal Distortions

In what follows, we will draw heavily from two publications: J. E. Tohline & I. Hachisu (1988, ApJ, 361, 394) — hereafter, TH88 — and K. Hadley & J. N. Imamura (2011a, Astrophysics and Space Science, 334, 1) — hereafter, HI11a.

Adopted Notation

Beginning with equation (2) of TH88 but ignoring variations in the vertical coordinate direction, the mass density is given by the expression,

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

Drawing from figure 2 of TH88, our Figure 1, immediately below, illustrates how the behavior of each factor in this expression can reveal itself during a numerical simulation that follows the time-evolutionary development of an unstable, nonaxisymmetric eigenmode. The initial model for this depicted evolution (model O3 from Table 1 of TH88) is a zero-mass — that is, it is a Papaloizou-Pringle like torus — with polytropic index,<math>~n = 3</math>, and a rotation-law profile defined by uniform specific angular momentum.

The top-left panel shows how, at any radial location, the phase angle, <math>~\phi_1/(2\pi)</math>, for the <math>~m=1</math> eigenmode, varies with time, <math>~t/t_\mathrm{rot}</math>, where, <math>~t_\mathrm{rot} \equiv 2\pi/\Omega_0</math> is the rotation period at the density maximum;
Using a semi-log plot, the top-right panel shows the exponential growth of the amplitude of three separate modes: The dominant unstable mode, displaying the largest amplitude, is <math>~m = 1</math>.
Using a semi-log plot (log amplitude versus fractional radius, <math>~\varpi/r_+</math>), the bottom-left panel displays the shape of the eigenfunction, <math>~f_1(\varpi)</math>, for the unstable, <math>~m=1</math> mode;
The bottom-right panel displays the radial dependence of the equatorial-plane phase angle, <math>~\phi_1(\varpi)</math>, for the unstable, <math>~m=1</math> mode; this is what TI11a refer to as the "constant phase loci."

Figure 1

Four panels extracted^† from figure 2, 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

^†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.

Empirical Construction of Eigenvector

Recently, while studying the series of three papers that was published by Hadley & Imamura in 2011, I was particularly drawn to the pair of plots presented in Figure 6 — and, again, in the top portion of Figure 13 — of HI11a. This pair of plots has been reprinted here, without modification, as our Figure 2. As in the bottom two panels of our Figure 1, the curves delineated by the blue dots in this pair of plots display (on the left) the shape of the eigenfunction, <math>~f_1(\varpi)</math>, and (on the right) the "constant phase loci," <math>~\phi_1(\varpi)</math>, for an unstable, <math>~m=1</math> mode. In this case, the initial model for the depicted evolution (the equilibrium model from Table 2 of HI11a having <math>~T/|W| = 0.253</math>) is a fully self-gravitating torus with polytropic index,<math>~n = 3/2</math>, and a rotation-law profile defined by a "Keplerian" angular velocity profile.

Figure 2

Panel pair extracted^† without modification from the top-most segment of Figure 13, p. 12 of K. Hadley & J. N. Imamura (2011a)

"Nonaxisymmetric Instabilities of Self-Gravitating Disks. I Toroids"

Astrophysics and Space Science, 334, 1 - 26 © Springer Science+Business Media B.V.

^†This pair of plots also appears, by itself, as Figure 6 on p. 12 of K. Hadley & J. N. Imamura (2011a).

Radial Eigenfunction

It occurred to me, first, that the blue curve displayed in the left-hand panel of HI11a's figure 6 (our Figure 2) might be reasonably well approximated by piecing together a pair of arc-hyperbolic-tangent (ATANH) functions. In an effort to demonstrate this, I began by specifying a "midway" radial location, <math>~r_- < r_\mathrm{mid} < r_+ \, ,</math> at which the two ATANH functions meet and the density fluctuation is smallest. Then I defined 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>

This empirically specified, two-piece <math>~f(\varpi)</math> function has been plotted in the left-hand panel of Figure 3; blue dots trace the function's behavior over the lower radial-coordinate range while green dots trace its behavior over the upper radial-coordinate range. It closely resembles the <math>~f(\varpi)</math> function (see the left-hand panel of our Figure 2) that developed spontaneously via HI11a's linear stability analysis.

Figure 3: Our Empirically Constructed Eigenvector

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

for

Hence, for the innermost region of the toroidal configuration — that is, over the lower radial-coordinate range — 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>~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr) </math>

for

and, similarly, we find that, over the upper radial-coordinate range,

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

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

for

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

Constant Phase Loci

Now let's work on the phase function, <math>~\phi_1(\varpi)</math>. The phase function displayed in the right-hand panel of our Figure 2 — that is, the phase function that developed spontaneously from HI11a's linear stability analysis — appears to be fairly constant (i.e., the phase is independent of radius) in the innermost region of the torus and, then again, fairly constant in the outermost region of the torus with a smooth but fairly rapid phase shift of approximately <math>~\pi</math> radians between the two extremes. This is the behavior exhibited by an arctangent (ATAN) function. With this in mind, we defined a new function, <math>~D(\varpi)</math>, in terms of the radial eigenfunction, <math>~f(\varpi)</math>, as follows:

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

<math>~\frac{f(\varpi) - f_\mathrm{min}}{f_\mathrm{max} - f_\mathrm{min}} \, .</math>

This has the following behavior:

At the inner edge of the torus <math>~(r_-)</math>, where <math>~f(\varpi) = f_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>;
At <math>~r_\mathrm{mid}</math>, where <math>~f(\varpi) = f_\mathrm{min}</math>, <math>~D(\varpi) = 0</math>;
At the outer edge of the torus <math>~(r_+)</math>, where again <math>~f(\varpi) = f_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>.

If we multiply <math>~D(\varpi)</math> by a sufficiently large number (we settled on "8" here), then it can serve as an argument of the ATAN function and satisfactorily swing the phase by <math>~\pi/2</math> over the inner (blue) region of the torus, then satisfactorily swing the phase by an additional <math>~\pi/2</math> over the outer (green) region of the torus.

Then we set,

<math>~\phi(\varpi) + \phi_0</math>

<math>~\biggl\{\tan^{-1}[8\cdot D(\varpi)] - \frac{\pi}{2} \biggr\} + \frac{\pi}{10} \, .</math>

Now, for the specific case being graphically illustrated here, <math>~f_\mathrm{min} = -2.99448</math> and <math>~f_\mathrm{max} = 2.64665</math>. Hence,

<math>~\phi(\varpi) + \frac{\pi}{2} </math>	<math>~=</math>	<math>~\tan^{-1}\biggl[8\cdot \biggl(\frac{f(\varpi) - f_\mathrm{min}}{f_\mathrm{max} - f_\mathrm{min}} \biggr)\biggr] </math>
	<math>~=</math>	<math>~\tan^{-1}[a\cdot f(\varpi) + b] \, , </math>

where, <math>~a = 1.41816</math> and <math>~b = 4.24664</math>.

User:Tohline/Appendix/Ramblings/Azimuthal Distortions

Contents

Analyzing Azimuthal Distortions

Adopted Notation

Empirical Construction of Eigenvector

Radial Eigenfunction

Constant Phase Loci

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