On the nonlinear saturation of the magnetorotational instability near threshold in a thin-gap Taylor-Couette setup

Abstract

We study the saturation near threshold of the axisymmetric magnetorotational instability (MRI) of a viscous, resistive, incompressible fluid in a thin-gap Taylor-Couette configuration. A vertical magnetic field, Keplerian shear and no-slip, conducting radial boundary conditions are adopted. The weakly non-linear theory leads to a real Ginzburg-Landau equation for the disturbance amplitude, like in our previous idealized analysis. For small magnetic Prandtl number (P \ll 1), the saturation amplitude scales as P^{2/3} while the magnitude of angular momentum transport scales as P^{4/3}. The difference with the previous scalings (~P^{1/2} and P respectively) is attributed to the emergence of radial boundary layers. Away from those, steady-state non-linear saturation is achieved through a modest reduction in the destabilizing shear. These results will be useful to understand MRI laboratory experiments and associated numerical simulations.

arXiv commentAccepted with minor changes to Physical Review E

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• Anomalous Low States and Long Term Variability in the Black Hole Binary LMC X-3
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• Numerical Study of the Magnetorotational Instability in Princeton MRI Experiment
• The weakly nonlinear magnetorotational instability in a local geometry
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• The weakly nonlinear magnetorotational instability in a global, cylindrical Taylor-Couette flow
• Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor-Couette flow
• Low magnetic-Prandtl number flow configurations for cold astrophysical disk models: speculation and analysis
• Hydrodynamical activity in thin accretion disks
• The magneto-rotational instability near threshold: spatio-temporal amplitude equation and saturation