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# Magnetorotational Instability

The magnetorotational instability or MRI (also known as the Velikhov instability or Balbus-Hawley instability in the literature) is a fluid instability that arises when the angular velocity of a magnetized fluid decreases as the distance from the rotation center increases. The MRI is of particular relevance in astrophysics where it is an important part of the dynamics in accretion discs.

A rotating fluid disc (e.g. an accretion disc) will remain in a laminar flow state as long as the angular momentum (per unit mass) increases outwards. This is also known as the Rayleigh stability criterion:

$\frac{\partial(r^2\Omega)}{\partial r}>0$

where Ω is the angular velocity of a fluid element and r is its distance to the rotation center. Instabilities leading ultimately to turbulence occur if a magnetic field is present and the angular velocity decreases with radius:

$\frac{\partial\Omega}{\partial r}<0$

Most accretion discs will meet this criterion.

The MRI was first noticed in a non-astrophysical context by Evgeny Velikhov in 1959 when considering the stability of Couette flow of an ideal hydromagnetic fluid.[1] His result was later generalized by S. Chandrasekhar in 1960.[2] More than 30 years later these early results were fully appreciated when S. A. Balbus and J. F. Hawley established that weak magnetic fields can substantially alter the stability character of accretion discs.[3]

A simple model illustrates the main effect of the instability. Consider a rotating fluid disc in the presence of a weak axial magnetic field. Two radially neighboring fluid elements behave as two mass points connected by a massless spring, the spring tension playing the role of the magnetic tension. In a Keplerian disc the inner fluid element orbits more rapidly than the outer, causing the spring to stretch. The inner fluid element is then forced by the spring to slow down, reduce correspondingly its angular momentum, and therefore move to a lower orbit. The outer fluid element, meanwhile, is forced by the spring to speed up, increase correspondingly its angular momentum, and therefore move to a higher orbit. The spring tension increases as the two fluid elements grow further apart, and eventually the process runs away.[4]

The observed accretion rates in astrophysical objects cannot be explained by a molecular viscosity, as the outward angular momentum transport in that case would not be enough to account for the inward flow of mass. The MRI provides a mechanism to account for the additional outward angular momentum transport. It is intrinsically a magnetohydrodynamic (MHD) phenomenon, with no hydrodynamic analog. Dynamo action is usually invoked to justify the existence of the magnetic field required for the instability to set in.[5]

Considerable effort has been put into realizing the MRI in a controlled laboratory experiment.[6][7] Results from these experiments will also allow testing of the accuracy of computer simulations involving the MRI.

## References

1. ^ Velikhov, E. P. (1959), " ", J. Exptl. Theoret. Phys. 36: 1398-1404
2. ^ Chandrasekhar, S. (1960), " ", Proc. Natl. Acad. Sci. 46: 253-257,
3. ^ Balbus, Steven A. & Hawley, John F. (1991), " ", Astrophysical Journal 376: 214-233,
4. ^ Balbus, Steven A. (2003), " ", Annu. Rev. Astron. Astrophys. 41: 555-597,
5. ^ Rüdiger, Günther & Hollerbach, Rainer (2004), , Wiley-VCH, ISBN 3-527-40409-0
6. ^ Sisan, Daniel R.; Mujica, Nicolás & Tillotson, W. Andrew et al. (2004), " ", Physical Review Letters 93 (11),
7. ^ Stefani, Frank; Gundrum, Thomas & Gerbeth, Gunter et al. (2006), " ", Physical Review Letters 97 (18),