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

Abstract

We conduct a global, weakly nonlinear analysis of the magnetorotational instability (MRI) in a Taylor–Couette flow. This is a multiscale, perturbative treatment of the nonideal, axisymmetric MRI near threshold, subject to realistic radial boundary conditions and cylindrical geometry. We analyze both the standard MRI, initialized by a constant vertical background magnetic field, and the helical MRI, with an azimuthal background field component. This is the first weakly nonlinear analysis of the MRI in a global Taylor–Couette geometry, as well as the first weakly nonlinear analysis of the helical MRI. We find that the evolution of the amplitude of the standard MRI is described by a real Ginzburg–Landau equation (GLE), whereas the amplitude of the helical MRI takes the form of a complex GLE. This suggests that the saturated state of the helical MRI may itself be unstable on long spatial and temporal scales.

Authors:
 [1];  [2]
  1. Department of Astronomy, Columbia University, New York, NY 10027 (United States)
  2. Department of Physics and Astronomy, Bates College, Lewiston, ME 04240 (United States)
Publication Date:
OSTI Identifier:
22663599
Resource Type:
Journal Article
Resource Relation:
Journal Name: Astrophysical Journal; Journal Volume: 841; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCRETION DISKS; AMPLITUDES; AXIAL SYMMETRY; BOUNDARY CONDITIONS; COUETTE FLOW; CYLINDRICAL CONFIGURATION; EVOLUTION; GINZBURG-LANDAU THEORY; INSTABILITY; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; NMR IMAGING; NONLINEAR PROBLEMS; PLASMA; SIMULATION

Citation Formats

Clark, S. E., and Oishi, Jeffrey S., E-mail: seclark@astro.columbia.edu. The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow. United States: N. p., 2017. Web. doi:10.3847/1538-4357/AA6FF6.
Clark, S. E., & Oishi, Jeffrey S., E-mail: seclark@astro.columbia.edu. The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow. United States. doi:10.3847/1538-4357/AA6FF6.
Clark, S. E., and Oishi, Jeffrey S., E-mail: seclark@astro.columbia.edu. Sat . "The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow". United States. doi:10.3847/1538-4357/AA6FF6.
@article{osti_22663599,
title = {The Weakly Nonlinear Magnetorotational Instability in a Global, Cylindrical Taylor–Couette Flow},
author = {Clark, S. E. and Oishi, Jeffrey S., E-mail: seclark@astro.columbia.edu},
abstractNote = {We conduct a global, weakly nonlinear analysis of the magnetorotational instability (MRI) in a Taylor–Couette flow. This is a multiscale, perturbative treatment of the nonideal, axisymmetric MRI near threshold, subject to realistic radial boundary conditions and cylindrical geometry. We analyze both the standard MRI, initialized by a constant vertical background magnetic field, and the helical MRI, with an azimuthal background field component. This is the first weakly nonlinear analysis of the MRI in a global Taylor–Couette geometry, as well as the first weakly nonlinear analysis of the helical MRI. We find that the evolution of the amplitude of the standard MRI is described by a real Ginzburg–Landau equation (GLE), whereas the amplitude of the helical MRI takes the form of a complex GLE. This suggests that the saturated state of the helical MRI may itself be unstable on long spatial and temporal scales.},
doi = {10.3847/1538-4357/AA6FF6},
journal = {Astrophysical Journal},
number = 1,
volume = 841,
place = {United States},
year = {Sat May 20 00:00:00 EDT 2017},
month = {Sat May 20 00:00:00 EDT 2017}
}
  • Hollerbach and Ruediger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this 'helical' MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only), and that it might be relevant to cooler astrophysical disks, especially those around protostars, which may be quite resistive. We confirm previous results for marginal stability and calculate HMRI growth rates. We show that in the resistive limit, HMRImore » is a weakly destabilized inertial oscillation propagating in a unique direction along the axis. But we report other features of HMRI that make it less attractive for experiments and for resistive astrophysical disks. Large axial currents are required. More fundamentally, instability of highly resistive flow is peculiar to infinitely long or periodic cylinders: finite cylinders with insulating endcaps are shown to be stable in this limit, at least if viscosity is neglected. Also, Keplerian rotation profiles are stable in the resistive limit regardless of axial boundary conditions. Nevertheless, the addition of a toroidal field lowers thresholds for instability even in finite cylinders.« less
  • Liu et al. [Phys. Rev. E 74, 056302 (2006)] have presented a WKB analysis of the helical magnetorotational instability (HMRI), and claim that it does not exist for Keplerian rotation profiles. We show that, if radial boundary conditions are included, the HMRI can exist even for rotation profiles as flat as Keplerian, provided only that at least one of the boundaries is sufficiently conducting.
  • We report experimental observation of an instability in a Couette-Taylor flow of a polymer fluid in a thin gap between two coaxially rotating cylinders in a regime where their angular velocity decreases with the radius while the specific angular momentum increases with the radius. In the considered regime, neither the inertial Rayleigh instability nor the purely elastic instability is possible. We propose that the observed 'elastorotational' instability is an analog of the magnetorotational instability which plays a fundamental role in astrophysical Keplerian accretion disks.