Hawley and Krolik: Toroidal Field

Hawley & Krolik: Simulations of the Plunging Region

Title Page | 3. Initially Poloidal Magnetic Field | 5. Discussion

4. Initially Toroidal Field

The second simulation begins with the same initial torus as before, but with a purely toroidal magnetic field. The most appropriate field topology in accretion disks is not known a priori and this simulation investigates how much the results depend on the initial field structure. We believe (as we discuss at greater length below) initially toroidal and initially poloidal fields bracket the range of possibilities.

The initial field configuration has toroidal field with a plasma $\beta=10$ wherever $\rho \ge 0.1$ . The model was run at two resolutions. One was the same as described above with 256 x 64 x 192 zones. This disk was evolved for 800,000 timesteps to time t=2704 (124 orbits at r_ms). The other, a lower resolution comparison simulation, is discussed below in §5.1.

The toroidal field simulation evolves in significantly different ways from the initially poloidal field case. First, with no initial radial field there is no shear amplification of the toroidal component during the early stages of the evolution. Second, the most unstable wavenumbers of the linear toroidal field MRI are different from those of the vertical field. For a pure toroidal field the fastest growth rates correspond to azimuthal wavenumber $m \sim v_{\phi}/v_A$ (i.e., m is large for typical subthermal field strengths) and large poloidal wavenumbers (Balbus & Hawley 1992; Terquem & Papaloizou 1996; Kim & Ostriker 2000). In previous local and global simulations done with a toroidal field (Hawley, Gammie, & Balbus 1995; H00), small scale fluctuations appear first, followed gradually by larger scale turbulence. As a result of these effects, the torus evolves more slowly than with a poloidal field, and the field energy and stress levels at saturation are smaller. In the present simulation the period of linear growth and evolution occupies the first $\sim 1000$ time units.

Nonetheless, the evolution of the disk after $\sim 1000$ units is in many ways qualitatively similar to the initially-poloidal simulation. Angular momentum is transported outward and its distribution evolves toward Keplerian. Smaller field energies and stresses cause the toroidal disk to be thinner than the poloidal field disk, with H/R remaining comparable to the initial value 0.12 at R=10 and $H/R \simeq 0.1$ inside R=5. Just as for the initially-poloidal simulation, there is substantial accretion through the inner cylindrical boundary and hardly any mass loss through the outer boundaries.

From $\sim 1000$ time units to the end, the accretion rate ranges between $\simeq 0.5$ and 2.5, i.e., between 8% and 40% of the accretion rate in the initially poloidal simulation (fig. 10). In contrast to the accretion rate history of the poloidal simulation, $\dot M$ builds slowly and features relatively smooth swings from a ``low state" to a ``high state" and back again. The ratio between the high and low accretion rates can be as large as $\sim 3$ ; there is no time at which $\dot M$ could fairly be said to be approximately stationary. In the poloidal case there are also large fluctuations between high and low rates, but these occurred on shorter timescales. Here there seems also to be a secular increase in $\dot M$ after t=1000, but its significance is hard to gauge as any long-term trend is masked by the very large fluctuations that occur. Figure 10 is the Fourier power s pectrum of the initially-toroidal accretion rate. It strongly resembles the power spectrum of the initially poloidal simulation, both with regard to its broken power-law character and the existence of a small peak (which in this simulation is at the slightly greater period of 47, corresponding to the circular orbit frequency at R=4.5). A second peak can be seen at a period of 142, which is the circular orbital frequency associated with R=8.7. The significance of these peaks is as uncertain as that of the similar peak found in the Fourier power spectrum of the initially-poloidal accretion rate.

Figure 10: Upper panel: Mass accretion rate at the inner edge as a function of time in the initially toroidal simulation. Lower panel: Fourier power density per logarithmic frequency interval of the accretion rate into the black hole (solid curve), and Fourier power density per logarithmic frequency interval of the volume-integrated Maxwell stress (dashed curve). To avoid transients associated with the initial start up and linear growth phase, the spectrum is computed for $t \ge 1000$ time units.

Because the accretion rate varies so much during this simulation, no single time can fairly represent its behavior. We will discuss two ``snapshots", one from a low accretion rate stage (at 2704 time units, when the accretion rate through the inner edge was $\simeq 1$ ), the other from a time of high accretion rate (2537 time units, when the accretion rate was 2.4).

In previous initial toroidal field simulations, both global and local, saturation occurs at lower turbulent field energies than models beginning with poloidal components. This is true in the present simulation as well. The azimuthally-averaged field strength (fig. 11) is about 10% stronger in the ``high rate" case compared to the low. Both, however, are about six times smaller than in the poloidal simulation when averaged over the accreting portion of the disk ( $R \leq 10$ , $\vert z\vert \leq 4$ ). In addition, the vertical scale height of the magnetic field is roughly half what it was in the initially poloidal case.

Figure 11: Azimuthally-averaged magnetic energy density in the ``high accretion rate" snapshot (upper panel) and the ``low rate" snapshot (lower panel). The scales in these two figures are identical to the scale used in fig. 2 to ease comparison.

Not surprisingly, given the generally weaker magnetic field, the effective $\alpha _{SS}$ is significantly smaller in the toroidal case than in the poloidal. In the poloidal simulation (fig. 4), $\alpha_{SS} \simeq 0.1$ at R=10 and rises sharply inward to a peak of almost 10 at the inner boundary. In the two toroidal snapshots, $\alpha_{SS} \simeq 0.01$ - 0.03 near R=10, and, although rising inward, it reaches a maximum of only $\simeq 0.2$ -0.3 just inside the marginally stable orbit. This contrast, of course, accounts for the lower accretion rate in the toroidal case, despite having initially an identical mass surface density. The value of $\alpha _{mag}$ , the ratio of the magnetic stress to the magnetic pressure, is 0.2-0.3 in the disk, rising from that value to $\sim 1$ from R=4 to the event horizon. This is similar to the poloidal field case, except that the systematic increase begins at a slightly smaller radius.

Comparatively weaker magnetic field also leads to a different shape to the stress distributions (fig. 12). The radial pressure gradient is much larger in the toroidal field case than in the poloidal field simulation: the vertically-integrated pressure falls by roughly a factor of 30 from R=10 to R=3, whereas the decline was only about a factor of 3 in the initially poloidal run. The magnetic stress, which rose steadily inward in the poloidal case, is approximately flat here between R=15 and R=5 and falls by about a factor of two from R=5 to $R \simeq 2$ -3, inside of which it is again constant.

Figure 12: Vertically-integrated and azimuthally-averaged pressure and magnetic stress at late times in the high and low accretion rate snapshots from the initially toroidal simulation. The solid curve shows the R- $\phi$ magnetic stress in the high accretion rate case; the dotted curve shows the pressure at the same time. The dashed and dot-dashed curves show the magnetic stress and pressure, respectively, in the low accretion rate snapshot.

Although the magnetic stress is generally weaker in the toroidal simulation than in the poloidal one, the mean change in specific angular momentum inside the marginally stable orbit is similar to that found in the poloidal simulation: a drop of 5 - 10% (fig. 13). On the other hand, the detailed character of the angular momentum distribution in the plunging region is quite different in the initially toroidal simulation. As we have already remarked, the change in energy and angular momentum of individual fluid elements is much greater than the mean change. When the accretion rate is especially high in the toroidal simulation, the contrast in specific angular momentum between adjacent fluid elements is much greater than when the accretion rate is low (fig. 14). Instead of passing through the inner boundary with specific angular momentum 2.6 (the angular momentum of the marginally stable orbit), in the high accretion rate case there are streams arriving with as little as $\simeq 1.8$ and some that arrive with as much as $\simeq 3$ . By contrast, during the time of low accretion rate, the range is only from $\simeq 2.2$ - 2.6.

The energy of accreting matter behaves in similar fashion, but with some notable contrasts. At the time of high accretion rate, the mean binding energy increases gradually from R=3 to R=2 to a maximum that is about 10% greater than at the marginally stable orbit, while at the time of low accretion rate there is little change in mean binding energy in the plunging region. However, just as in the poloidal simulation, the slow change in mean energy masks very large changes in the energy of individual fluid elements (fig. 15). As with the angular momentum, t he energy exchange inside R=2 is much stronger in the case of initially toroidal field: whereas the maximum increase in binding energy in the poloidal case was about a factor of two, in the toroidal case it is a factor of 10 - 20! Finally, in all cases, there is a spike in the mean binding energy just outside R_min that is an artifact of the boundary c ondition.

Figure 13: Mass flux weighted azimuthally- and vertically-averaged specific binding energy and angular momentum as functions of radius in the initially toroidal simulation. The solid curve shows binding energy in the high accretion rate state, the dotted curve angular momentum; the dashed and dash-dotted curves are energy and angular momentum, respectively, in the low accretion rate state. All four curves are normalized to their values at R=3 to emphasize the continuing change at smaller radii. The spikes in the binding energy just outside the inner radius of the simulation are artifacts.

Figure 14: Mass-weighted vertically-averaged specific angular momentum in the inner part of the accretion flow for the initially toroidal case: upper panel, at high accretion rate; lower panel, at low accretion rate.

Figure 15: Net energy in rest-mass units in a slice through the equatorial plane in the plunging region during the time of high accretion rate in the initially toroidal simulation.

Title Page | 3. Initially Poloidal Magnetic Field | 5. Discussion