Inductionless Magnetorotational Instabilities: From Lab Tests To Accretion Disks

How stars and black holes are able to form from rotating matter is one of the big questions of astrophysics. What is known is that magnetic fields figure prominently into the picture via the mechanism of magnetorotational instability (MRI). However, the current understanding is that it only works if matter is electrically well conductive – but in rotating disks this is not always the case. In areas of low conductivity like the dead zones of protoplanetary disks or the far-off regions of accretion disks that surround supermassive black holes, the MRI’s effect is numerically difficult to comprehend and is thus a matter of dispute. Simulation of the MRI in a liquid metal experiment with an exclusively vertically oriented magnetic field requires that this field has to be rather strong. At the same time, since the rotational speed has to be very high, these types of experiments are extremely involved and thus far success has eluded them. By adding a circular magnetic field to a vertical one it became possible to observe the helical MRI at substantially smaller magnetic fields and rotational speeds. Very recently, the azimuthal MRI with m=1 has also been observed in the PROMISE facility in Dresden. However, one of the blemishes of these inductionless versions of MRI is the fact that they only act to destabilize rotational profiles that are relatively precipitous towards the periphery, which for now did not include rotation profiles obeying Kepler’s law. In this talk we present a study of the stability of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence characterized by an appropriate magnetic Rossby number. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive case, our main focus is on the limit of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We rigorously demonstrate that the inductionless versions of MRI extend well to the Keplerian case if only the azimuthal field slightly deviates from its field-free profile.