Hawley: 3.2 Results

3. Results

3.2 Disk with initial vertical field

The two general classes of models considered in the local shearing box limit by HGB95 were initial vertical fields and initial toroidal fields. In the next simulation, CK6, we complete the extension of these local models from the shearing box to the cylindrical disk by simulating the same disk as CK5, but now with an initial vertical magnetic field.

Simulation CK6 is a Keplerian disk consisting of an isothermal gas at constant density $\rho = 1$ from R=10 to the outer boundary, and a constant sound speed c_s = 0.03054. The radial grid resolution is the same as in CK5, but a few additional zones are added in the $\phi$ and z direction; $\phi$ runs from 0 to $\pi/2$ over 64 grid zones, and z runs from 0 to 2 in 32 grid zones. In CK6 the strength of the initial vertical magnetic field is set by requiring $2\pi v_A/\Omega=0.5$ at every radius. This is nearly the fastest growing wavelength of the linear MRI; there will be four such wavelengths across the z domain. The initial magnetic field decreases with radius (proportional to $\Omega$ ), with values of $\beta$ ranging from 240 at the inner edge of the disk to 66,000 at the outer grid boundary. Although the most unstable vertical wavelength is the same throughout the disk, the local growth rate of the MRI is proportional to the local orbital frequency, so the magnetic instability grows more slowly with increasing radius.

The simulation is run out to time t=4097 which is 22.9 orbits at the disk's initial inner boundary (R= 10), 188 orbits at r_ms, and only 1.37 orbits at the grid outer boundary. As in CK5, MHD turbulence develops first in the inner disk and spreads outward. In CK6 the instability grows more rapidly, and although the field is much weaker initially, it still grows to about the same level as in CK5.

In CK6 accretion is driven through the marginally stable orbit at early time. This is due, in part, to a unique property of the vertical field instability. Large fluctuations and impulsive accretion are created by the ``channel solution'' of the nonlinear vertical field instability (Hawley & Balbus 1992; HGB95). The fastest growing modes have finite vertical wavenumber k_z and vanishing radial and azimuthal wavenumbers, k_R and m. This leads to coherent radial flow upon nonlinear saturation. Although the development of full turbulence eventually limits the coherence of this radial channel flow, early on in the simulation at the inner edge of the disk radial filaments of gas are able to move rapidly inward and accrete.

As the simulation proceeds, an interesting structure emerges in the inner part of the disk. By t=2000 gas has piled up in a density maximum located at R=7.5 and in a smaller maximum at R=4. Between these rings are local minima at R=5 and R=10. These can be seen in Figure 3, a spacetime diagram of the vertically- and azimuthally-averaged density. For clarity, only the inner half of grid is shown and only up to t=2000. A local maximum in density corresponds to a local minimum in the Maxwell stress and the Alfvén speed and vice versa. As time proceeds, the inner density maximum accretes into the hole, while the other slowly drifts inward, reaching R=4.2 by t=4096. By this time another density gap is opening up near R=17.

Figure 3: (R,t) spacetime diagram of the vertically and azimuthally averaged density ${\langle \rho \rangle}$ in simulation CK6 for the inner half of the radial grid from t=0 to 2000. The disk is not initially in hydrostatic equilibrium at the inner edge, resulting in an outward traveling wave after t=0. The turbulence itself subsequently generates magnetoacoustic waves. Of particular interest are the local density maxima and the gaps surrounding them which develop after t=1000 at R=5 and R=10.

The specific angular momentum, density, and Maxwell stress at the end time are shown in Figure 4. The Maxwell stress, and hence the angular momentum transport, is larger in regions of low density. The radial slope of the specific angular momentum $\ell$ becomes steeper in the regions of maximum Maxwell stress, and flatter in the density maxima. Essentially, the disk is forming a local pressure-supported slender torus with non-Keplerian angular momentum distribution to balance the internal pressure forces. Gas moves rapidly out of the region between the rings, but is blocked from further accretion by the torus itself. Thus there is a tendency for material to concentrate into rings.

Figure 4: Vertically- and azimuthally-averaged values at t=4096 in simulation CK6. (a) Specific angular momentum $\langle \ell\rangle$ overlaid on the Keplerian angular momentum (dashed line), (b) density ${\langle \rho \rangle}$ , and (c) Maxwell stress normalized by the initial gas pressure, $M^{R\phi }/\rho c_s^2$ .

The presence of dense rings and empty gaps is a description of the behavior of a ``viscous-type'' instability (Lightman & Eardley 1974) where the stress is a decreasing function of the density. Here the cause of this behavior is different from that considered originally by Lightman and Eardley (1974). It is not a property of the Maxwell stress per se as much as it is a consequence of the properties of the MRI. Lower density corresponds to larger Alfvén speeds which, in turn, produce faster growth rates at larger wavelengths of the MRI. Longer wavelengths are more efficacious in transporting larger amounts of angular momentum.

A spacetime plot of the accretion rate shows evidence for significant time variability and periodicity. The variability is manifest as pressure waves that propagate throughout the disk at the sound speed. A fourier transform of the spacetime data reveals peaks in the power spectrum which originate at various radii in the disk. The peaks occur where the pressure waves are generated at frequencies close to the local orbital frequency. Initially the largest peak in the spectral energy is found close to R=13; this is also where the peak Maxwell stress is generated. At late time the most prominent frequency observed corresponds to the orbital frequency at the location of the dense inner ring. This, in turn, drives a periodic accretion flux into the central hole.

The substantial spatial inhomogeneity of the disk at late time makes it difficult to assign overall ``steady state'' values to quantities. The azimuthal density fluctuations, characterized by (8) are as large as 0.3 in the region of maximum Maxwell stress, and are between 0.1 and 0.2 throughout the bulk of the radial extent of the disk. This level is comparable to that seen in the toroidal field simulation CK5. What sets CK6 apart are the large variations with radius.

The magnetic field has undergone substantial amplification. At the end of the run $\beta$ is about 70 inside the dense ring at R=4, and falls to 0.2 in the low density region at R=7. From R=10 to 40, $\beta$ has a minimum of 14, with a local maximum $\sim 70$ at r=20. Beyond R=30 $\beta$ slowly rises with radius to around 100 at the outer boundary. As stated above, the Maxwell stress is largest between the dense rings. The Shakura-Sunyaev $\alpha$ is as low as 0.002 in the inner ring, rising to $\sim 4$ at R=7 before dropping back to values ranging from 0.01 to 0.05 between R=10 and 40. As in run CK5 the average total toroidal field energy dominates over that of the poloidal field, although here by a lesser amount, with $B_\phi^2/B_R^2 = 6.3$ . The radial field energy is a factor of 6 greater than the vertical field energy. The ratio of the Maxwell stress to the magnetic pressure, $\alpha_{mag}$ , varies with radius, but has an average of 0.5. This is larger than seen in CK5, and is the typical value for local shearing box simulations with a net vertical field.

Title Page | 3.1 Disk with initial toroidal field | 3.3 Disk near the marginally-stable orbit