Harvard Astronomy 201b

The Magnetorotational Instability

In Uncategorized on April 11, 2011 at 12:53 am

Introduction

The MRI is an instability that arises from the action of the magnetic field in a differentially rotating system (i.e. a disk), and can lead to large scale mixing and turbulence very quickly (MRI grows on a dynamical timescale t_{MRI} \propto 1/\Omega where \Omega is the rotational frequency). The necessary conditions for the MRI to develop are the following:

  • There is a weak poloidal magnetic field (i.e. the field points in a direction normal to the disk)
  • The disk rotates differentially, where \frac{d\Omega}{dR} < 0

Given that magnetic fields are ubiquitous and that astrophysical disks (which rotate differentially) are commonplace, the MRI arises in a huge diversity of astrophysical settings (including X-ray binaries, the Galactic disk, and protoplanetary disks).

 

Motivation + Historical Context

A longstanding problem in astrophysics has been to explain the large observed viscosities in accretion disks (inferred from binary systems with high accretion rates).  Molecular viscosity alone was far too weak, as were other mixing mechanisms such as convection and tidal mixing.  It was concluded that a turbulent mixing engine was needed to generate the observed degree of viscosity – in other words, an instability was needed to drive the mixing!  However in the absence of magnetic fields (this simplifying assumption was made since in many situations, magnetic fields tend to resist change, and it was thought that the action of magnetic fields would only stabilize the system), no instabilities could be found that would drive such turbulence in a disk!

A breakthrough came with the rediscovery of the MRI (Balbus and Hawley 1991), whose dynamics were originally studied in the 1950’s by Chandrasekhar (1953, 1961) and independently by Velikhov (1959).   Balbus and Hawley demonstrated that the MRI does indeed manifest itself in accretion disks, which could account for the turbulent mixing needed to explain the observed mass accretion rates.

 

What Causes the MRI?

Figure 1: Motion of the fluid parcels (red and blue circles) carries the magnetic field lines (purple line) along with them. The resultant magnetic forces are shown (red and blue arrows) which arise due to tension in the field lines.

Magnetic Fields as Springs:

 
In an ideal plasma (i.e. a magnetized, perfectly conducting fluid), the action of the magnetic field is to link neighboring fluid parcels that lie along a common field line.  One property of ideal plasmas is that magnetic field lines are frozen in the fluid parcels – the motion of the fluid carries the magnetic field along with it!  This lends itself to a beautifully simple physical picture of the magnetic forces; the fluid parcels can be thought of a beads tied together on a string.  If for some reason fluid parcels start to diverge, the tension in the magnetic “string” acts to bring the connected fluid elements back together (see Figure 1).

Intuitively, we can just think of the magnetic force acting on the fluid as springs tying neighboring fluid elements together!  Normally, we think of a spring as a restoring force, which tends to preserve a system’s stability.  However, if the fluid motion occurs in a differentially rotating frame, it turns out that this restoring force can actually lead to the system destabilizing!  This is the origin of the MRI.

The action of differential rotation and magnetic field line tension

Figure 2 : Growth of the MRI due to differential rotation

 

A Consequence of Differential Rotation:

 
Figure 2 (right) illustrates how the instability arises.  Consider the following mechanical model with two neighboring fluid parcels that are connected by a spring (representing magnetic force) in a differentially rotating disk.  The inner parcel (blue) rotates faster than the outer parcel (red).  The resulting force acting on the fluid elements depends only on their separation.  This leads to the following feedback cycle (which is the essence of the MRI):

  • a) Magnetically connected fluid elements have some initial displacement
  • b) Differential rotation increases the displacement, and magnetic tension causes the inner parcel to slow down, and the outer parcel to speed up
  • c) This transfer of angular momentum causes the inner parcel to migrate inwards and the outer parcel to be pushed outwards.
  • d) Repeat from step a, but now with a larger displacement.

However, if the magnetic field is too strong (i.e. the tension in the spring is too strong), the feedback cycle won’t run.  The tension will instead cause the displacement between fluid parcels to oscillate rather than grow in step b).  For a more detailed analysis, please refer to Balbus and Hawley 1991.

 

Numerical Simulations

Modelling the interplay between magnetic fields acting on a differentially rotating plasma is in general a highly complex nonlinear problem that is only tractable through simulations.  Numerical experiments have been set up in the following ways to simulate the growth of MRI on various scales (see links for movies):

From the simulations, we find that the flow resulting from MRI is inherently turbulent rather than viscous.   The growth of the MRI is independent of the initial field strength, however the field strength does affect the size scale of the strongest growing modes.  It also appears that the MRI acts an an effective dynamo (which strengthens and aligns the magnetic field in the disk). This dynamo action leads to the launching of jets and other outflows that escape perpendicular to the disk.

 

Links for Further Reading:

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