Analyzing Azimuthal Distortions

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,

 $~\rho$ $~=$ $~\rho_0 \biggl[ 1 + f(\varpi)e^{-i(\omega t - m\phi)} \biggr] \, ,$

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

Using the subscript, $~m$, 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,

 $~\omega_m$ $~=$ $~\omega_R + i\omega_I \, ,$

we expect each unstable mode to display the following behavior:

 $~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]$ $~=$ $~f_m(\varpi)e^{-i[\omega_R t + i \omega_I t - m\phi_m(\varpi)]}$ $~=$ $~\biggl\{ f_m(\varpi)e^{-im\phi_m(\varpi)}\biggr\} e^{-i\omega_R t } \cdot e^{\omega_I t}$ $~=$ $~\biggl\{ f_m(\varpi)e^{-i[\omega_R t + m\phi_m(\varpi)]} \biggr\} e^{\omega_I t} \, .$

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

 $~y_1 \equiv \frac{\omega_R}{\Omega_0} - m$ and $~y_2 \equiv \frac{\omega_I}{\Omega_0} \, ,$

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

 $~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]$ $~=$ $~\biggl\{ f_m(\varpi)e^{-i[(y_1+m) (\Omega_0 t) + m\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)}$ $~=$ $~\biggl\{ f_m(\varpi)e^{-im[(y_1/m+1) (\Omega_0 t) + \phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, .$

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

$\Omega_p \equiv \frac{\omega_R}{m} = \Omega_0\biggl(\frac{y_1}{m}+1\biggr) \, ,$

we should find an eigenvector of the form,

 $~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]_\mathrm{rot} \equiv \biggl[ \frac{\rho}{\rho_0} - 1 \biggr]e^{im\Omega_p t}$ $~=$ $~\biggl\{ f_m(\varpi)e^{-im[\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)} \, ,$

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 †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, $~r_- < r_\mathrm{mid} < r_+ \, ,$ at which the density fluctuation is smallest. Then define a function of the form,

 $~f(\varpi)$ $~=$ $~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) \biggr]$ for $r_- < \varpi < r_\mathrm{mid} \, ;$ and $~f(\varpi)$ $~=$ $~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr) \biggr]$ for $r_\mathrm{mid} < \varpi < r_+ \, .$

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

 PRACTICAL IMPLEMENTATION:   At the two limits, $~\varpi = r_-$ and $~\varpi = r_+$, the function, $~f(\varpi) \rightarrow +\infty$; while, at the limit, $~\varpi = r_\mathrm{mid}$, the function, $~f(\varpi) \rightarrow -\infty$. In practice we stay half of a radial zone away from these three limiting radial boundaries, so that the maximum and minimum values of $~f(\varpi)$ 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:

 $~\tanh^{-1}x$ $~=$ $~\frac{1}{2} \ln\biggl( \frac{1+x}{1-x} \biggr)$ for $x^2 < 1 \, .$

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

 $~x$ $~=$ $~1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)$ $~\Rightarrow ~~~~ \frac{1+x}{1-x}$ $~=$ $~\biggl[2 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr] \biggl[2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]^{-1}$ $~=$ $~[(r_\mathrm{mid}-r_-) - ( \varpi - r_-)] [(\varpi - r_-)]^{-1}$ $~=$ $~\frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \, .$

Therefore we can write,

 $~f(\varpi)$ $~=$ $~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr)$ for $r_- < \varpi < r_\mathrm{mid} \, ;$

and, similarly, we find,

 $~f(\varpi)$ $~=$ $~\frac{1}{2} \ln\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_+} \biggr)$ for $r_\mathrm{mid} < \varpi < r_+ \, .$