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Reduced Modeling of the Magnetorotatational Instability



Accretion Disks


Accretion disks are are pervasive in astrophysics being found in protoplanetary disks, the merger of two solar masses, and around black holes in galactic nuclei. However, accretion disks require a mechanism for efficient angular momentum exctraction. Hydrodynamical processes are unable to produce the required angular momentum transfer to allow for the existence of these disks.
The Magnetorotational Instability is thought to provide this increased angular momentum transfer by generating turbulence and increasing the effective viscosity in the disk.


Reduced Model


Assuming that the instability favors small vertical wavenumbers, and that the dissipative processes are small relative to angular velocity and shear, an asymptotic analysis provides the first order equations of motion for the evolution and saturation of this instability. A key feature of the asymptotics is allowing the local shear stress to feed back onto the shear imposed by differential rotation.

This video shos the growth of the instability in this reduced model and the transition to saturation. Here we allow for a vertical magnetic field (illustrated by contour lines of phi) and added a small random perturbation. The boundary conditions are periodic in the vertical and impenetrable stress-free in the horizontal. The critical wavenumber from the linear theory grows until the nonlinear terms dominate. The flow eventually saturates, as the local shear stress (V' in this movie) reaches a constant value and coarsens to the lowest wavenumber supported in the computational domain.