Who carries angular momentum through an accretion disk?

Picture two narrow rings of gas around a star. The inner ring completes an orbit sooner. Yet a kilogram in the outer ring carries more angular momentum: its larger distance from the centre outweighs its slower orbital speed. Angular momentum measures this combination of sideways motion and distance from the axis. Gas that moves inward along a sequence of nearly circular orbits must give some of it to something else.

A displaced parcel normally comes back

Give a parcel a small outward push without giving it a sideways torque. Its angular momentum stays nearly fixed. At its new radius, it has less angular momentum than the gas already on a circular orbit there. Gravity turns it back inward. In a Keplerian disk, where the central mass dominates gravity, this motion is a small oscillation around the original orbit.

This explains part of the puzzle. Rotation that gets slower farther out supplies a large store of energy, but a small displacement does not automatically draw on it. Ordinary molecular viscosity is often too weak to explain the transport required in astrophysical disks. Something must connect the displacement to a sustained torque.

A bent field links the two motions

Now imagine a weak magnetic field threading electrically conducting gas. The ideal-conductor approximation lets the gas move with the field. When neighbouring portions move apart, the field bends. A bent magnetic field exerts tension, a force that tends to straighten it. The field is not a material string; the string picture helps us follow the direction of this magnetic force.

The faster inner portion pulls ahead of the outer portion. Magnetic tension slows the inner gas and speeds up the outer gas. The inner portion loses angular momentum and moves inward; the outer portion gains angular momentum and moves outward. That increases their difference in orbital motion and bends the field further. A suitable small disturbance therefore reinforces itself.

Balbus and Hawley's 1991 linear analysis showed how this feedback operates in weakly magnetized disks. Their component equations connect radial motion, azimuthal motion and the corresponding magnetic disturbances. In the unstratified vertical-field case considered here, angular velocity must decrease outward and magnetic tension must lie in a growing range for the chosen wavelength. A very strong connection makes the disturbance oscillate. At a fixed wavelength, an ever weaker field eventually gives slower growth too.

Motion and magnetic tension both transport momentum

Outward-moving gas can carry extra forward momentum, and inward-moving gas can carry a deficit. This correlation transports angular momentum outward even if the average radial motion of the disturbance is zero. Physicists call its momentum-flux contribution the Reynolds stress. Here, stress means momentum crossing a surface per unit area and time.

Magnetic tension supplies a second contribution, called the Maxwell stress. In a growing channel, the radial and azimuthal magnetic disturbances have opposite signs. Their tension pulls forward on the outer side and backward on the inner side. These signs make the magnetic contribution to outward transport positive.

The saved linear example in the figure makes the two contributions concrete. The magnetic stress is about five-thirds of the velocity stress, so roughly five-eighths of the total comes from the magnetic field. This ratio belongs to this particular fastest-growing Keplerian mode. Other disturbances have different ratios, and their stresses can vary or change sign.

The energy comes from differential rotation, the difference in angular speed between neighbouring radii. The disturbance grows as it takes energy from that background flow.

A growing channel is the start of the story

The accompanying saved calculation follows a small, vertically periodic disturbance. Its velocity and magnetic variations grow together; the gas moves horizontally, with different motion at different heights. “Vertical channel” names the direction in which the pattern varies. It does not describe gas streaming vertically out of the disk.

One local time unit is the inverse orbital angular speed; a full orbit takes about 6.28 of these units. In three local time units, the radial velocity amplitude in the saved fastest-mode example grows by about 9.49 times. The calculation ends while perturbations are still small and the linear approximation applies. A real disk's later transport requires additional physics and nonlinear evolution. The linear channel explains the initial feedback and the direction of momentum transfer; it supplies no turbulence saturation level or global accretion rate.

Try the mechanism on a new case. If both rings rotate at the same angular speed, does a magnetic connection still draw energy from their differential rotation? There is no such differential rotation to tap. Magnetic forces can exchange energy between motion and field, but this source of exponential growth is absent. That comparison is the starting point for the local equations.

Sources

• S. A. Balbus and J. F. Hawley, A Powerful Local Shear Instability in Weakly Magnetized Disks. I. Linear Analysis, ApJ 376,214–222 (1991), especially printed pages 216–218, equations 2.2,2.9 and2.13–2.14b. Primary scan.
• J. M. Stone and T. A. Gardiner, Implementation of the Shearing Box Approximation in Athena, arXiv:1006.0139v1,1 June 2010, Sections I–II. Pinned primary source.

The numerical bar and amplitude example use the preserved keplerian-fastest linear spatial-channel record; the source and plotting method are documented in the figure source notes. They are simulated linear fields.