3. Results

3.1 Disk with initial toroidal field

The first Keplerian disk simulation, CK5, considers a disk with an inner edge at R=10, set back from rms. This will investigate disk accretion removed from the dynamical forces associated with the pseudo-Newtonian potential's innermost circular orbit. The simulation follows the global development of the MRI, turbulent angular momentum transport, and the resulting gradual accretion of matter and inward drift of the disk edge.

The initial conditions consist of an isothermal gas at constant density, $\rho = 1$, and a sound speed cs = 0.03054 which gives a Mach number ${\mathcal M} = v_\phi/c_s = 11.5$ at R=10. With the decrease in velocity $v_\phi$ the Mach number declines to 4.3 by the outer radial boundary. The azimuthal angle $\phi $ runs from 0 to $\pi/2$ in 58 grid zones, and in the vertical direction z runs from 0 to 2 in 24 grid zones. The vertical scale height at R=10 corresponds to 0.87. Of course in the cylindrical disk approximation the absence of a vertical gravitational force means that the scale height has no hydrodynamical significance in z; it is merely a convenient expression of the temperature of the disk in terms of the ratio of the sound speed to orbital frequency.

With a magnetic field, the vertical domain size retains significance even in the cylindrical disk limit in relation to the most unstable wavelength of the MRI. The magnetic field strength determines this wavelength, and the initial pressure determines the value of $P/P_{mag} \equiv \beta$ to which this field strength corresponds. For CK5 the initial magnetic field is purely toroidal with $\beta = 4$. For this strength of field the ratio of the Alfvén speed to the orbital frequency, $v_A/\Omega$, ranges from 0.6 at the inner disk boundary to 10.2 at R=60. The critical azimuthal wavenumber for the MRI, $m=R\Omega/v_A$, ranges from 16 at the inner disk to 6 at the outer boundary. Since the azimuthal direction spans $\pi/2$ with 58 grid zones, only wavenumbers $4 \le m \le 116$ will be realizable. With the chosen magnetic field strength, however, the unstable azimuthal wavenumbers of the MRI should be well resolved throughout the disk.

The simulation is run out to time t=3972 which is 22 orbits at the disk's initial inner boundary, and 180 orbits at rms. As the simulation proceeds the MRI develops, first at the inner disk edge and spreading outward through the disk as time advances. Small wavelength perturbations (large m) grow most rapidly, but the largest azimuthal scales dominate in the end. The exponential growth of the poloidal field at a given radius ends after approximately five orbits, as measured locally. The Maxwell stress in the resulting MHD turbulence transports angular momentum and the disk evolves. Figure 1 shows a spacetime (R,t) plot of the Maxwell stress, $M^{R\phi }$, in terms of a Shakura & Sunyaev (1973) $\alpha$ using the initial pressure as a normalization, $\alpha =M^{R\phi }/\rho c_s^2$. This plot illustrates the rapid local buildup of magnetic stress to significant levels, as well as the large fluctuations in time and space.

Figure 1: Spacetime diagram of the Maxwell stress in run CK5 normalized by the initial pressure $\alpha =M^{R\phi }/\rho c_s^2$. The stress develops first at R=10 and then spreads out through the disk as the MRI grows at a rate proportional to the local orbital frequency. At the end of the simulation the stress rises rapidly from near zero at R=5, reaches the highest levels between R=15 and 30, then declines rapidly with radius. Within the disk the stress varies strongly in both time and space.


Overall, the disk never attains a steady state; its inner edge slowly moves inward and fluid in the outer disk moves outward. Through the course of the simulation the disk inner edge moves from R=10 down to R=5. This is illustrated in Figure 2 which shows ${\langle \rho \rangle}$ at several times during the run. At the end time, the disk has not quite reached rms and there is no significant mass flux through the marginally stable orbit and into the central hole. Magnetic field has been carried inward along with the mass. The regions with the strongest fields lie between the peaks in the density distribution. The average $\beta$ value varies between 8 at the density peaks and 2 in between.

Figure 2: Radial profile of density ${\langle \rho \rangle}$ at times t=0 (solid bold line), 890 (short dashed line), 1835 (dash-dotted line), 2734 (thick dashed line), 3848 (solid thin line) in simulation CK5. These times correspond to 0, 4.9, 10.2, 15.2, and 21.5 orbits at R=10. Angular momentum transport produced by MHD turbulence causes the inner edge of the disk to drift gradually inward.


Averaged properties of the accretion flow within the disk can be derived from a region between R=13 and 22 which comes into an approximate time-averaged steady state after t=2800. The time-averaged accretion rate within this region is roughly constant with radius, with $\dot M = 0.075$, and $v_R/v_\phi = -0.006$. The instantaneous fluctuations in radial velocity, $\delta v_R/v_\phi$ typically range between 0.01 and -0.02. Within the turbulent disk the toroidal field energy dominates over that of the poloidal field, with the ratio $B_\phi^2/B_R^2 = 13$. The radial field energy is, in turn, a factor of 6 greater than the that of the vertical field. The toroidal field energy is essentially unchanged in the mean from its initial value, although there are spatial variations in strength by a factor of two within the disk. At the end of the run the region of significant Maxwell stress has moved out to around R=40 with the largest values lying between R=7 and 30 (Fig. 1). The time-averaged Maxwell stress over this range is about 5.5 x 10-5, corresponding to an average value of $\alpha = 0.07$. The relatively low level of the radial field compared to the toroidal is reflected in the value of $\alpha_{mag}$, the ratio of the Maxwell stress to the total magnetic pressure ( $\alpha = \alpha_{mag}/\beta$). In the inner disk this value ranges between 0.2 and 0.3. The toroidal field simulation of ARC obtained an $\alpha$ slightly below 0.01, and an $\alpha_{mag} \sim 0.3$. (Note that there are differences in $\alpha$ terminology between this paper and ARC.)

The averaged magnetic properties of the turbulence are consistent with the results of toroidal field local shearing box simulations (cf. Table 3 of HGB95). For those simulations $\alpha$ was typically a few times 0.01, and $\alpha_{mag} \sim 0.4$ for simulations that began with weak toroidal fields and $\sim 0.2$ for those that began, like this simulation, with stronger toroidal fields.

Title Title Page   |   Method 2. Problem Setup   |   section3.2 3.2 Disk with initial vertical field