structure: midplane (x-y) slice. Magnetic pressure is shown along with projected magnetic field vectors. The inner half of this picture corresponds to Fig. 2. [See the electronic edition of the Journal for a color version of this figure.] Fig. 2.—Run of supporting stresses (radial components of momentum fluxes), averaged on radial shells and compared to the local gravitational force per unit volume. Also plotted is the standard deviation of gravitational force per unit volume on radial shells. [See the electronic edition of the Journal for a color version of this figure.] box edge. The simulations ran successfully for 6000 time steps to , or 1.5 free-fall times from rB . At that point, mag- t p 650 netic fields squeezed out along the midplane to the outer bound- ary, leading to numerical instabilities. With a radial resolution of 102.85 zones, of which 101.51 is used for central and outer boundary conditions, we have 1.34 decades of scale in which to arrange the Kepler radius rK and the magnetic turnaround radius . We performed runs that are initially ro- rmag tationally supported ( ) and purely rotationally supported r ! r mag K ones ( ), as well as magnetized but nonrotating initial r p 0 mag conditions ( ). Fluid is initialized with solid-body rotation r p 0 K on shells and constant specific angular momentum on cones of constant polar angle. The production simulation (Fig. 1) chose , . To further break discrete symmetries, r p 336 r p 100 K mag velocity was modulated at the 5% level at multipoles up to . l p 2 These simulations differ from previous ones in two important ways: (1) magnetic and viscous dissipation occur only at the grid scale (or inner boundary); no enhanced diffusivity or vis- cosity is applied; and (2) we break the alignment usually as- sumed between magnetic and rotational axes. This was accom- plished with large-scale flux loops, misaligned to the rotational axis, that are dragged toward the central object. 2.1. Results Simulations were run about one dynamical time at rB . Al- though this does not give us any handle on the long-term evo- lution of the flow, it does yield a picture of how the inner flow responds—for many inner dynamical times—to initial and boundary conditions. A snapshot of the resulting magnetic field structure is shown in Figure 1, and various shell-averaged radial stresses are plotted in Figure 2. At this point in the simulation, a central hydrostatic region supported by gas pressure—not rotation or magnetic fields— has grown outward. Within the region plotted in Figure 2, for , also, (so that , Ϫn Ϫ1.51 Ϫ0.79 r ∝ r n Ӎ 0.72 P ∝ r T ∝ r whereas was expected), and magnetic pressure Ϫ1 r P ∼ mag . Gas pressure gradients dominate magnetic stresses by Ϫ1.5 10 P a factor of ∼10; Reynolds stresses, including rotation and in- flow, are smaller still and switch direction. Rotation is roughly one-tenth Keplerian, despite the fact that the entire region is a factor of 2 inside the initial rK . The ratio of Alfve ´n to inflow velocities is similarly ∼10, indicative of magnetic braking. In contrast to Bondi and ADAF-type flows, inflow is very subsonic; correspondingly, the mass accretion rate is strongly suppressed relative to Bondi’s estimate. However, the accretion rate does agree with the Bondi rate derived from conditions at the inner boundary. This state is not rotationally supported like the CDAF. It resembles in some ways the convection- dominated Bondi flow (CDBF) of IN02 and Gruzinov (2001), but see § 3 for differences. To test the role of magnetic fields in the clogged inflow, we suddenly turned the magnetic fields off in the nonrotating scenario and evolved the fluid for a dynamical time. The flow returned to Bondi’s solution. 3. PHYSICAL INTERPRETATION The inner region plotted in Figure 2 is in quasi-hydrostatic equilibrium with a polytropic index . Since the ad- g Ӎ 2.25 eff iabatic index is , this represents a strongly super- g p 5/3 adiabatic state. In the usual description of entropy-driven con- vection, this can occur only when convective velocities approach the sound speed, which in a power-law atmosphere is roughly free-fall. Saturation at a constant Mach number is a feature of Quataert & Gruzinov’s (2000) CDAF model and the CDBF model of IN02, both of which have and fall 1 n p 2 within class II of Gruzinov’s (2001) classification of self- similar nonradiative flows. This slope is flat enough to satisfy MAGNETICALLY FRUSTRATED CONVECTION L209 cture: midplane (x-y) slice. Magnetic pressure magnetic field vectors. The inner half of this See the electronic edition of the Journal for a Fig. 2.—Run of supporting stresses (radial components of momentum fluxes), averaged on radial shells and compared to the local gravitational force per unit volume. Also plotted is the standard deviation of gravitational force per unit volume on radial shells. [See the electronic edition of the Journal for a color version of this figure.] ns ran successfully for 6000 time steps all times from rB . At that point, mag- along the midplane to the outer bound- l instabilities. n of 102.85 zones, of which 101.51 is used dary conditions, we have 1.34 decades ge the Kepler radius rK and the magnetic responds—for many inner dynamical times—to initial and boundary conditions. A snapshot of the resulting magnetic field structure is shown in Figure 1, and various shell-averaged radial stresses are plotted in Figure 2. At this point in the simulation, a central hydrostatic region supported by gas pressure—not rotation or magnetic fields— has grown outward. Within the region plotted in Figure 2, for , also, (so that , Ϫn Ϫ1.51 Ϫ0.79 r ∝ r n Ӎ 0.72 P ∝ r T ∝ r whereas was expected), and magnetic pressure Ϫ1 r P ∼ mag . Gas pressure gradients dominate magnetic stresses by Ϫ1.5 10 P a factor of ∼10; Reynolds stresses, including rotation and in- flow, are smaller still and switch direction. Rotation is roughly one-tenth Keplerian, despite the fact that the entire region is a factor of 2 inside the initial rK . The ratio of Alfve ´n to inflow velocities is similarly ∼10, indicative of magnetic braking. In contrast to Bondi and ADAF-type flows, inflow is very subsonic; correspondingly, the mass accretion rate is strongly suppressed relative to Bondi’s estimate. However, the accretion 364 J. Cuadra et al. Pen et al. 2003: In MHD, w/ ordered strong eld, accretion is ~spherical, even with rotation! (also Pang et al. 2011) Cuadra et al. 2006: In hydro, the gas cools, circularizes and forms a disk Saturday, September 1, 2012
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