The Origin of the Slow-to-Alfv\'en Wave Cascade Power Ratio and its Implications for Particle Heating in Accretion Flows

Abstract

The partition of turbulent heating between ions and electrons in radiatively inefficient accretion flows plays a crucial role in determining the observational appearance of accreting black holes. Modeling this partition is, however, a challenging problem because of the large scale separation between the macroscopic scales at which energy is injected by turbulence and the microscopic ones at which it is dissipated into heat. Recent studies of particle heating from collisionless damping of turbulent energy have shown that the partition of energy between ions and electrons is dictated by the ratio of the energy injected into the slow and Alfv\'en wave cascades as well as the plasma $\beta$ parameter. In this paper, we study the mechanism of the injection of turbulent energy into slow- and Alfv\'en- wave cascades in magnetized shear flows. We show that this ratio depends on the particular ($r\phi$) components of the Maxwell and Reynolds stress tensors that cause the transport of angular momentum, the shearing rate, and the orientation of the mean magnetic field with respect to the shear. We then use numerical magnetohydrodynamic shearing-box simulations with background conditions relevant to black hole accretion disks to compute the magnitudes of the stress tensors for turbulence driven by the magneto-rotational instability and derive the injection power ratio between slow and Alfv\'en wave cascades. We use these results to formulate a local subgrid model for the ion-to-electron heating ratio that depends on the macroscopic characteristics of the accretion flow.