Interaction of the Magnetorotational Instability with Hydrodynamic Turbulence in Accretion Disks

Abstract

Accretion disks in which angular momentum transport is dominated by the magnetorotational instability ( MRI ) can also possess additional, purely hydrodynamic, drivers of turbulence. Even when the hydrodynamic processes, on their own, generate negligible levels of transport, they may still affect the evolution of the disk via their influence on the MRI. Here we study the interaction between the MRI and hydrodynamic turbulence using local MRI simulations that include hydrodynamic forcing. As expected, we find that hydrodynamic forcing is generally negligible if it yields a saturated kinetic energy density that is small compared to the value generated by the MRI. For stronger hydrodynamic forcing levels, we find that hydrodynamic turbulence modifies transport, with the effect varying depending on the spatial scale of hydrodynamic driving. Large-scale forcing boosts transport by an amount that is approximately linear in the forcing strength, and leaves the character of the MRI (e.g., the ratio between Maxwell and Reynolds stresses) unchanged, up to the point at which the forced turbulence is an order of magnitude stronger than that generated by the MRI. Low-amplitude small-scale forcing may modestly suppress the MRI. We conclude that the impact of hydrodynamic turbulence on the MRI is generically ignorable in cases, such as convection, where the additional turbulence arises due to the accretion energy liberated by the MRI itself. Hydrodynamic turbulence may affect (and either enhance or suppress) the MRI if it is both strong and driven by independent mechanisms such as self-gravity, supernovae, or solid-gas interactions in multiphase protoplanetary disks.

Influence of Ohmic diffusion on the excitation and dynamics of MRI

“…For these parameters our analysis predicts roughly a factor of 30 difference in the limiting rms-values, the exact numbers reading u rms /c s = 0.6 for the large-scale forcing, and u rms /c s = 0.0225 for the small-scale one. Workman & Armitage (2008) report a running time average of the kinetic energy density E kin = 0.0012 for the pure MRI case, and roughly 15 times more kinetic energy for the largest forcing amplitude investigated. Assuming that the average density is roughly one, the rms-velocities would read u rms /c s ≈ 0.049 and 0.190, for the pure and maximally forced runs, respectively.…”

“…The results of Snellman et al (2009) indicate that in this regime the stress R xy is small and negative although the runs are probably not directly comparable due to the different forcing scales. The conclusion of Workman & Armitage (2008) that hydrodynamic stresses are small is based on this one point, which just by chance lies in an unfortunate part of the parameter space.…”

“…Let us now consider the implications of the above analysis to the earlier numerical work done by Workman & Armitage (2008). They used a setup very similar to the one presented in Sect.…”

“…11. In the hydrodynamic run pre- sented by Workman & Armitage (2008) the Coriolis number, Co = 2Ω/(u rms k f ), measuring the effect of rotation, is roughly 5.3. The results of Snellman et al (2009) indicate that in this regime the stress R xy is small and negative although the runs are probably not directly comparable due to the different forcing scales.…”

Quantifying the effect of turbulent magnetic diffusion on the growth rate of the magneto−rotational instability

“…Following earlier work of Workman & Armitage (2008) and Korpi et al (2010), we solve the three-dimensional equations of magnetohydrodynamics (MHD) in a cubic domain of size L 3 in the presence of rotation with angular velocity Ω = (0, 0, Ω), a shear flow U S = (0, S x, 0) with shear S = − 3 2 Ω, and an imposed magnetic field B 0 = (0, 0, B 0 ). We adopt shear-periodic boundary conditions in the x direction (Wisdom & Tremaine 1988) and periodic boundary conditions in the y and z directions.…”