Saturation of Magnetorotational Instability Through Magnetic Field Generation

Abstract

The saturation mechanism of Magneto-Rotational Instability (MRI) is examined through analytical quasilinear theory and through nonlinear computation of a single mode in a rotating disk. We find that large-scale magnetic field is generated through the alpha effect (the correlated product of velocity and magnetic field fluctuations) and causes the MRI mode to saturate. If the large-scale plasma flow is allowed to evolve, the mode can also saturate through its flow relaxation. In astrophysical plasmas, for which the flow cannot relax because of gravitational constraints, the mode saturates through field generation only.available in accretion disks and other astrophysical settings because of strong gravitational constraints that maintain a Keplerian profile (V φ ∝ r −1/2 ). Instead, the MRI may saturate (cease linear growth) if the non-linear evolution of the instability generates a mean component of the magnetic field that is sufficient to stabilize the mode.We remark that for most plasma instabilities, if the energy source is fixed and unvarying (and nonlinear coupling to other modes is ignored), the mode will grow without bound, until limited by dissipation or some physical dimension of the system. In this sense, the MRI might be atypical.In this paper we investigate the linear growth and non-linear saturation mechanism and amplitude of a single MRI mode in thick-disk geometry, i.e., L r ∼ 1. Our primary tool is linear and non-linear MHD computation in cylindrical (r, φ, z) geometry which is periodic in the azimuthal (φ) and axial (z) directiona. We solve a model initial value problem in which the inner and outer radial boundaries are impermeable, perfectly conducting, concentric cylinders that can rotate independently at specified rates. They are coupled to the internal flow by hydrodynamic viscosity. The initial mean (azimuthally and axially averaged) profile V φ (r) is Keplerian, and in most cases is assumed to be maintained for all times by external forces. Perturbations to the equilibrium that depend on (r, φ, z) are then introduced, and evolve dynamically according to the single fluid MHD equations. Linear growth rates and eigenfunctions are found by integrating the linearized, single fluid, visco-resistive MHD equations forward in time until an exponentially growing solution is obtained. The saturation mechanism and amplitude of the mode are determined by solving the non-linear MHD equations for a single mode, with azimuthal mode number m and axial mode number k, beginning from a small initial perturbation. At finite amplitude, the mode will interact with itself to modify the mean background state, which in turn alters both its temporal evolution and its radial dependence. In this sense it differs from a quasi-linear calculation,

Radially dependent large-scale dynamos in global cylindrical shear flows and the local cartesian limit

“…The amplification of Bz from axisymmetric modes (Eq. 23), which may contribute to the mode saturation in a cylinder (Ebrahimi et al 2009) results from the curvature terms and is absent for channel modes in a local Cartesian model, as discussed below.…”

“…However due to toroidal symmetry, B φ = 0 as the positive and negative contributions from the perturbations remain on the same vertical surface, and only the mean vertical field is amplified through ∂E φ /dr. (Ebrahimi et al 2009). …”

“…Largescale fields in MRI flows have been associated with the sustenance of MRI turbulence (Lesur & Ogilvie 2008;Davis et al 2010;Simon et al 2011) are correlated with the convergence of Maxwell stress (Guan & Gammie 2011;Nauman & Blackman 2014), and can influence corona formation (Blackman & Pessah 2009). For a single MRI mode, large-scale magnetic fields generated via an EMF can cause MRI saturation (Ebrahimi et al 2009). In short, the large-scale dynamos of MRI-unstable systems are of interest both as phenomena on their own, and because they may be closely connected to angular momentum transport in accretion disks by local and nonlocal Maxwell stresses (Blackman & Nauman 2015).…”

The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow

“…The MRI feeds off of the free energy from differential rotation, and so a modification of the background shear may cause saturation (Knobloch & Julien 2005;Umurhan et al 2007b). The MRI may transfer its free energy into the magnetic field, and saturate when the field is too strong to be susceptible to the MRI (Ebrahimi et al 2009). The MRI may saturate differently depending on the particular parameter regime under investigation, and so our challenge is not only in identifying possible saturation mechanisms, but in understanding how and when each applies in different astrophysical environments.…”

Magnetorotational instability: recent developments

“…One possibility is that the energy extracted by the MRI goes into amplifying the magnetic field, although perhaps only on a resistive time scale (Ebrahimi et al 2009). However, no simulations can reach the large Reynolds numbers for which the asymptotic approach described here is designed.…”