3. Results

3.4 Influence of phi domain

One of the potentially most useful approximations for three-dimensional simulations is to reduce the azimuthal angular coverage to some integer fraction of $2\pi$. The advantage of this is obvious: CK6 covered $\pi/2$ in 64 grid zones, whereas CK7 required 256 grid zones to span the full $2\pi$ with the same $\Delta \phi$. Model CK7a investigates the potential drawbacks of this reduction in computational domain size by providing a direct comparison between a simulation spanning $\pi/2$ and one spanning the full $2\pi$ (CK7). The grid size $\Delta \phi$ is the same in both simulations.

The two simulations are very similar in their qualitative appearance, although there are quantitative differences. CK7a has about 10% less magnetic energy and magnetic stress on average. On the other hand, CK7a exhibits larger fluctuations in those quantities. These differences carry over into the accretion rate into the black hole. In CK7 $\dot M$ is 11% larger on average, but CK7a has larger impulsive spikes in accretion rate.

The larger fluctuation level seen in CK7a may be due, in part, to existence of the channel solution for vertical fields. Ideally the channels have a finite kz and kR = m = 0. However, parasitic instabilities (Goodman & Xu 1994) with nonzero m and kR < kz cause the breakup of the channel solution into smaller scale turbulence. Reducing the azimuthal extent of the computational domain apparently makes it easier for the channel solutions to maintain some spatial coherence for slightly longer time.

Does the reduced domain influence the amount of stress at the marginally stable orbit? The cylindrical disk simulations of ARC found a smaller decrease in specific angular momentum between rms and the inner boundary compared with the thick disk simulation of HK. Since ARC used an azimuthal domain of only $\pi/6$ in angle it is possible that some of the reduction in stress was due to the restricted angular domain. In CK7a the average $\alpha$ value at rms after t=1000 is 0.053, although $\alpha$ does briefly go as high as 0.15 at several points in time. The mean value of $\ell $ at rms is 2.64 and the decrease between that point and the inner boundary is 0.032, or 1.2%. This is less than seen in CK7 (see Fig. 6), and is consistent with the general reduction in average stress levels in the reduced domain simulation. Thus, although the smaller domain size reduces the stress, this effect plays only a minor role in the quantitative differences in the simulations of HK and ARC.

As with CK7, the MHD turbulence produces propagating pressure waves. These are generated primarily in the inner portion of the disk at frequencies corresponding to the orbital frequency (rather than the periodicity frequency $\Omega/4$). These waves are tightly wrapped, low m trailing spirals. Figure 8a compares an angular power spectrum of density $\rho$ for CK7 and CK7a at t=1500. The spectrum is averaged over radius from R=5 to 20. Both simulations show increasing power toward smaller azimuthal wavenumbers m. This is consistent with the visual appearance of the disk which in which low m spiral waves dominate. The steepness of the slope of the power spectrum increases with m, from $\sim -2$ at low m, up to $\sim -7$ for large m. The two simulations have very similar power spectra; CK7a is simply truncated by symmetry at m=4.


Figure 8: Azimuthal fourier power spectra of (a) the vertically-averaged density, and (b) the magnetic field components BR (lower solid curve), $B_\phi $ (top solid curve), and Bz (dashed curve). The data are taken from at the end time for CK7a (heavy curves) and the corresponding time in CK7, and averaged over space from R=5 to 20. Density peaks at the smallest m wavenumbers, while the magnetic power spectra have a break in the slope between m=10 and 20. In general, there is good qualitative agreement between the two simulations over their common angular domain.


The situation is somewhat different for the magnetic field. Figure 8b shows the azimuthal power spectra for the magnetic field components in runs CK7 and CK7a. What is most striking here is the break in the spectral slopes between m=10 and 20. The same phenomenon was noted by Armitage (1998; see Fig. 3) in a full $2\pi$ simulation of a cylindrical disk with an initial vertical field. These plots suggest CK7 and CK7a are so similar because the primary input from the MRI is on scales $\sim H$, as would be expected for a vertical field, and $H \ll R$. The inverse cascade to smaller m numbers leaves CK7 with roughly 10% of the energy at the largest azimuthal scales, but this doesn't significantly alter the evolution of the MRI.

The azimuthal domain size might be more crucial in simulations of disks containing an initial toroidal magnetic field rather than a vertical field. With toroidal fields the instability depends directly upon the azimuthal wavenumber m. For example, the most unstable wavenumber has $m/R v_A \approx \Omega$. For low m this corresponds to a field for which $v_A \sim v_\phi$, which would be an exceptionally strong field. Under such circumstances a full $2\pi$ global treatment would obviously be required. Even with weaker toroidal fields, when $v_A \ll v_\phi$, low m modes remain unstable, albeit with smaller growth rates. Further, in a shearing background, nonaxisymmetric waves have a time-dependent radial wavenumber. According to linear theory (Balbus & Hawley 1992; Terquem & Papaloizou 1996) for a given azimuthal mode m, maximum field amplification occurs when the the wavenumber quantity (k/kz)2 is small, hence field growth is maximized with low values kR and/or high values of kz. Since kR evolves as $mt d\Omega / dR$ a reduction in total amplification can occur if low m modes are removed by a reduced angular domain size.

For a test comparison, we examine two cylindrical disk simulations that were computed for a separate project. These simulations are labeled NK1 and NK1a and are listed in Table 1. Each begins with identical initial conditions except that NK1a uses a $\pi/2$ angular domain and NK1 the full $2\pi$. These simulations use a Newtonian gravitational potential with GM=1 on a grid that runs from R=0.25 to 3.75, and 0 to 0.2 in z. Note that this make the time units different from the other runs; the orbital period at the center of the grid (R=2) is 17.8. The initial condition consists of an isothermal Keplerian disk. The temperature profile is fixed so that the Mach number ${\mathcal M} = c_s/v_\phi = 20$ is constant with radius. The pressure is also constant, and the density increases as R. The initial magnetic field is toroidal and of strength $\beta = 4$. The most unstable wavenumber is $m\cong v_\phi/v_A= {\mathcal M}\sqrt{2 \beta} =57$; this is well above the m=4 limit of the $\pi/2$ grid.

Both simulations have a similar local growth rate for the perturbed magnetic field during the first few local orbital periods. Then the growth rate for NK1a drops off while NK1 continues unabated to saturation at a time of 10 local orbits. The field energy in NK1 is higher than in NK1a, but NK1a continues to grow with a slower rate, and manages to achieve about the same level as NK1 after 20 orbits. Beyond this point in time, the energies fluctuate around a mean, and the mean values differ by about 10%. Similarly, the mean magnetic stress in NK1a is 13% lower than in NK1.

An examination of the fourier power spectrum for the density and the magnetic field components in run NK1 shows that the power in the field components continues to rise until about m=4. Beyond, to smaller m values, the power is flat ($B_\phi $) or slightly decreasing (BR). In contrast, the power in CK7 turns over at larger m values. The difference is that when the background field is toroidal there is power at m=0. Further, the toroidal field is unstable for all nonzero m below the critical wavenumber $m= {\sqrt 3} v_\phi/v_A$.

To conclude, although there are significant differences in the initial growth stage of the MRI, and there is power in the lowest m modes, by the end of the run the reduced domain size has the same effect on the toroidal field simulations as for those with vertical field: a 10% reduction in averaged field strengths. The reduction of the $\phi $ domain is a good approximation because the weak-field MRI is essentially a local instability. The MHD turbulence is driven locally, and the qualitative behavior of the disk is unchanged so long as the field remains weak, $\beta > 1$.

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