Torus Diagnostics

The Kerr GR hydrodynamics code is used to study the evolution of the Papaloizou-Pringle instability in thick tori. In keeping with H91, the evolution of the instability is characterized by several reduced quantities extracted from the simulation data. These include the maximum density enhancement at mode saturation, $\delta \rho/\rho$, the Fourier power in density for the azimuthal wavenumbers, m=1 and 2, q_sat, the angular velocity distibution parameter at mode saturation, and the mass flux in the equatorial plane (to help shed light on the relationship between accretion and the growth of the PPI). In all models, the calculations are done 20 times per orbit at the initial pressure maximum. The only exception is the mass flux, which is computed once per orbit.

The density enhancement, $\delta \rho/\rho$, is obtained by finding the maximum density at in the data arrays, and computing $\delta \rho = \rho_{max}{\vert}_{t=t_{sat}}-\rho_{max}{\vert}_{t=0}$.

We extract the m=1 and m=2 Fourier modes by computing azimuthal averages using the numerical equivalents of

$$\Re{k}_m(r) \int_{0}^{2\pi}{\rho(r,\pi/2,\phi)\,\cos{(m\,\phi)}\,d \phi}$$ = (12)

$$\Im{k}_m(r) \int_{0}^{2\pi}{\rho(r,\pi/2,\phi)\,\sin{(m\,\phi)}\,d \phi}.$$ = (13)

The mode power is then

$$f_m ={1 \over r_{out}-r_{in}}\int_{r_{in}}^{r_{out}}{\log_e{\left({(\Re{k}_m(r))}^2 +{(\Im{k}_m(r))}^2\right)}\, d r}.$$ (14)

A linear fit is performed to the time-sequenced data to extract a mode growth rate. For this calculation r_in and r_out are the initial values for the inner and outer edges of the torus in the equatorial plane.

In H91, the parameter q_sat was obtained by a radial power law fit to the azimuthally averaged angular velocity, $\bar{\Omega}(r) \sim r^{-q}$, at mode saturation. This parameter was used to characterize deviations from a purely Keplerian profile. Since H91 dealt with Schwarzschild black holes, the angular velocity for the equilibrium fat disk had a simple form, $\Omega = U^\phi/U^t \simg_{tt}\,r^{-2}$, i.e. the usual Keplerian profile multiplied by the redshift factor. It was therefore straightforward to compare the final disk profile against the Keplerian case. With Kerr black holes, the equilibrium fat disk has a more complicated radial dependence, $\Omega = (g^{t \phi} - l g^{\phi\phi})/(g^{t t} - l g^{t \phi})$. However, since the aim is to measure a change in the equatorial angular velocity profile, we adapt the procedure. As with H91, we obtain the azimuthally averaged angular velocity using the numerical equivalent of

$$\bar{\Omega}(r) = {1\over 2\pi} \int_{0}^{2\pi}{V^{\phi}(r,\pi/2,\phi)\,d \phi}.$$ (15)

A power-law fit $\bar{\Omega}(r) \sim r^{-q}$ is obtained from the slope of a log-log plot of $\bar{\Omega}(r)$ for $r \in \left(r_{in},r_{out}\right)$ at the time step corresponding to mode saturation. We also extract the initial value q₀ in an analogous manner, and report the change $\delta\,q =q_{sat}-q_{0}$ as a measure of the redistribution of angular velocity.

The azimuthally-averaged mass flux in the equatorial plane is computed using the numerical equivalent of

$${\dot{M}}_{eq}(r) = {\sqrt{\gamma(r,\pi/2)}\over 2\pi}\int_{0}^{2\pi}{D(r,\pi/2,\phi)\,V^{\phi}(r,\pi/2,\phi)\,d \phi}.$$ (16)

In the figures below ${\dot{M}}_{eq}(r)$ is plotted at a radius r lying just inside the initial inner edge of the torus r_in and is used to establish the presence of a flow of matter towards the black hole.

2002-06-05