Magnetorotational Instability: Nonmodal Growth and the Relationship of Global Modes to the Shearing Box

Abstract

We study the magnetorotational instability (MRI) (Balbus & Hawley 1998) using nonmodal stability techniques. Despite the spectral instability of many forms of the MRI, this proves to be a natural method of analysis that is well-suited to deal with the non-self-adjoint nature of the linear MRI equations. We find that the fastest growing linear MRI structures on both local and global domains can look very different to the eigenmodes, invariably resembling waves shearing with the background flow (shear waves). In addition, such structures can grow many times faster than the least stable eigenmode over long time periods, and be localized in a completely different region of space. These ideas lead -for both axisymmetric and non-axisymmetric modes -to a natural connection between the global MRI and the local shearing box approximation. By illustrating that the fastest growing global structure is well described by the ordinary differential equations (ODEs) governing a single shear wave, we find that the shearing box is a very sensible approximation for the linear MRI, contrary to many previous claims. Since the shear wave ODEs are most naturally understood using nonmodal analysis techniques, we conclude by analyzing local MRI growth over finite time-scales using these methods. The strong growth over a wide range of wave-numbers suggests that nonmodal linear physics could be of fundamental importance in MRI turbulence (Squire & Bhattacharjee 2014).

Nonlinear Transverse Cascade and Sustenance of MRI Turbulence in Keplerian Disks with an Azimuthal Magnetic Field

“…In particular, in magnetized disks, the nonmodal/transient growth of the MRI over intermediate (dynamical) times can be actually more relevant in many situations than its modal growth (Mamatsashvili et al 2013;Squire & Bhattacharjee 2014). Since in the present case of azimuthal field, the MRI exhibits only transient rather than exponential growth, the resulting turbulence, like in spectrally stable HD shear flows, is expected to be governed by a subtle cooperation of this nonmodal growth and nonlinear processes.…”

“…In order to understand this dependence and hence appropriately select the aspect ratio in simulations, in our opinion, one should take into account as fully as possible the nonmodal growth of the MRI during intermediate (dynamical) timescales, because it can ultimately play an important role in the turbulence dynamics (Squire & Bhattacharjee 2014). However, this is often overlooked in numerical studies.…”

“…This approach has already been successfully applied to (magnetized) disk flows (Mukhopadhyay et al 2005;Zhuravlev & Razdoburdin 2014;Squire & Bhattacharjee 2014;Razdoburdin & Zhuravlev 2017). Such perturbations yield maximum linear nonmodal growth during finite times and therefore are responsible for most of the energy extraction from the background flow.…”

“…The corresponding initial conditions, leading to this highest growth at t d are called optimal perturbations. (A reader interested in the details of these calculations is referred to Squire & Bhattacharjee (2014), where the formalism of optimal growth and optimal perturbations in MRI-active disks, which is adopted here, is described to a greater extent.) Reference time, during which to calculate the nonmodal growth, is generally arbitrary.…”

“…To this aim, we apply the method of optimal perturbations, widely used in fluid dynamics for characterizing the nonmodal growth in spectrally stable shear flows (see e.g., Farrell & Ioannou 1996;Schmid & Henningson 2001;Zhuravlev & Razdoburdin 2014), to the present MRI problem (see also Squire & Bhattacharjee 2014). These are perturbations that undergo maximal transient growth during the dynamical time.…”