The Saturation Limit of the Magnetorotational Instability

Simulations of the magnetorotational instability (MRI) in a homogeneous shearing box have shown that the asymptotic strength of the magnetic field declines steeply with increasing resolution. Here I model the MRI driven dynamo as a large scale dynamo driven by the vertical magnetic helicity flux. This growth is balanced by large scale mixing driven by a secondary instability. The saturated magnetic energy density depends almost linearly on the vertical height of the typical eddies. The MRI can drive eddies with arbitrarily large vertical wavenumber, so the eddy thickness is either set by diffusive effects, by the magnetic tension of a large scale vertical field component, or by magnetic buoyancy effects. In homogeneous, zero magnetic flux, simulations only the first effect applies and the saturated limit of the dynamo is determined by explicit or numerical diffusion. The exact result depends on the numerical details, but is consistent with previous work, including the claim that the saturated field energy scales as the gas pressure to the one quarter power (which we interpret as an artifact of numerical dissipation). The magnetic energy density in a homogeneous shearing box will tend to zero as the resolution of the simulation increases, but this has no consequences for the dynamo or for angular momentum transport in real accretion disks. The claim that the saturated state depends on the magnetic Prandtl number may also be an artifact of simulations in which microphysical transport coefficients set the MRI eddy thickness. Finally, the efficiency of the MRI dynamo is a function of the ratio of the Alfv'en velocity to the product of the pressure scale height and the local shear. As this approaches unity from below the dynamo reaches maximum efficiency.

💡 Research Summary

The paper revisits the long‑standing puzzle that magnetorotational‑instability (MRI) simulations in a homogeneous shearing box show a dramatic decline of the saturated magnetic field strength as numerical resolution is increased. Rather than treating the MRI‑driven dynamo as a direct consequence of shear amplifying small‑scale fields, the author proposes a large‑scale dynamo model in which the key driver is the vertical magnetic helicity flux generated by the turbulent eddies. In the nonlinear stage of the MRI, each eddy expels a net amount of magnetic helicity in the vertical direction; because the box is periodic in the horizontal directions, this helicity flux cannot be cancelled locally and therefore builds up a mean vertical magnetic field.

Growth of the mean field is counteracted by a secondary instability that mixes the large‑scale field with the turbulent motions. This mixing can be thought of as a “large‑scale turbulence” whose characteristic vertical scale is the eddy thickness (h). By balancing the helicity‑driven growth rate against the mixing rate, the author derives a simple scaling for the saturated magnetic energy density:

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