Exploring non-normality in magnetohydrodynamic rotating shear flows: Application to astrophysical accretion disks

The emergence of turbulence in shear flows is a well-investigated field. Yet, there are some lingering issues that have not been sufficiently resolved. One of them is the apparent contradiction between the results of linear stability analysis showing a flow to be stable and yet experiments and simulations proving it to be otherwise. There is some success, in particular in astrophysical systems, based on magnetorotational instability (MRI), revealing turbulence. However, MRI requires the system to be weakly magnetized. Such instability is neither a feature of general magnetohydrodynamic (MHD) flows nor revealed in purely hydrodynamic flows. Nevertheless, linear perturbations of such flows are non-normal in nature, which argues for a possible origin of nonlinearity therein. The concept behind this is that non-normal perturbations could produce huge transient energy growth (TEG), which may lead to nonlinearity and further turbulence. However, so far, non-normal effects in shear flows have not been explored much in the presence of magnetic fields. In this spirit, here we consider the perturbed viscoresistive MHD shear flows with rotation in general. Basically we recast the magnetized momentum balance and associated equations into the magnetized version of Orr-Sommerfeld and Squire equations and their magnetic analogs. We also assume the flow to be incompressible and in the presence of Coriolis effect solve the equations using a pseudospectral eigenvalue approach. We investigate the possible emergence of instability and large TEG in three different types of flows, namely, the Keplerian flow, the Taylor-Couette (or constant angular momentum) flow, and plane Couette flow. We show that, above a certain value of magnetic field, instability and TEG both stop occurring. We also show that TEG is maximum in the vicinity of regions of instability in the wave number space for a given magnetic field and Reynolds number, leading to nonlinearity and plausible turbulence. Rotating shear flows are ubiquitous in astrophysics, especially accretion disks, where molecular viscosity is too low to account for observed data. The primary accepted cause of energy-momentum transport therein is turbulent viscosity. Hence, these results would have important implications in astrophysics.