Magnetorotational instability

 

The magnetorotational instability is widely believed to be responsible for outward angular momentum transport in astrophysical accretion disks. The efficiency of this transport depends on the amplitude of this instability in the saturated state. We employ an asymptotic expansion based on an explicit, astrophysically motivated timescale separation between the orbital period, Alfv\'en crossing time and viscous or resistive dissipation timescales, originally proposed by Knobloch and Julien, Phys. Fluids 17, 094106 (2005), to formulate a semi-analytical description of the saturated state in an incompressible disk. In our approach, a Keplerian shear flow is maintained by the central mass but the instability saturates via \rev{the generation of a mean vertical magnetic field}. The theory assumes that the time-averaged angular momentum flux and the radial magnetic flux are constant and determines both self-consistently. The results predict that, depending on parameters, steady saturation may be supercritical or subcritical, and in the latter case that the upper (lower) solution branch is always stable (unstable). The angular momentum flux is always outward, consistent with the presence of accretion, and for fixed wavenumber peaks in the subcritical regime. The limit of infinite Reynolds number at large but finite magnetic Reynolds number is also discussed. 

 

The right figure shows different regimes of saturation. Regime I denotes the supercritical regime containing an unstable trivial state and a stable finite amplitude state. Regime II denotes the subcritical region with three states, the stable trivial state, the upper branch nonlinear states, and the lower branch intermediate unstable nonlinear states. In regime III one finds an unstable trivial state and a stable but disconnected finite amplitude state.  

 

(Work with Prof. Keith Julien and Prof. Edgar Knobloch.)