Astron. Astrophys. 333, 603-612 (1998)

3. Angular momentum transport processes

3.1. internal gravity waves

The kinetic energy of convective motions in the envelope appears as a source of pressure fluctuations at the boundary between the envelope and the core. These fluctuations propagate as internal gravity waves through the core. The waves carry angular momentum and their dissipation therefore transmits torques. Assume that prograde and retrograde waves are excited with the same amplitude. This is a good approximation if the rotation is slow, such that the effect of Coriolis forces on the wave generation process is small. As the waves propagate into the core, they conserve their angular momentum (or wave action) until dissipation becomes important. If the dissipation of prograde and retrograde waves is the same, no net angular momentum transport takes place. Prograde (retrograde) waves propagating into a medium of increasing (decreasing) rotation speed, however, meet critical layers (where the rotation rate equals the horizontal component of the phase speed), and are much more effectively absorbed there (see the discussions in Goldreich & Nicholson 1989, Zahn et al. 1997). Due to this asymmetry, there is a net angular momentum transport which tends to reduce the differential rotation. In effect, the internal gravity wave field is a source of friction. Zahn et al. find that for the Sun, the time scale for synchronization between core and envelope due to this friction is of the order yr. Since this is of the order of evolution time scale to the giant branch, the process could be significant in maintaining synchronization during core contraction. Detailed evolution calculations including this process by Talon & Zahn (1998), however, still yield too large internal rotation rates for the present Sun.

3.2. magnetic torques

Magnetic torques are transmitted by the stress component . Approximating this as constant over a spherical surface, the angular momentum balance is

The synchronization time scale between a core of radius cm, mass and moment of inertia rotating at a rate s^-1 is then

where k is the gyration radius. At the Sun's current rotation rate , the synchronization time scale is less than the age of the Sun if G. If on its way to the giant branch the core of the Sun were to contract to a radius of cm while conserving angular momentum, it would rotate at rate . To maintain corotation on the yr contraction time scale, a field strength G is sufficient, at this rotation rate.

If the azimuthal and radial field components are of similar magnitude, (1) can be written in terms of the magnetic energy , and rotational energy as

If the and are not comparable, (3) is only the minimum magnetic energy required. Nevertheless, since is typically such a large number, (3) shows that corotation can be maintained by a magnetic energy which is small fraction of the rotational energy of the star, for spindown on a time scale long compared with the rotation period.

3.3. Winding up of field lines

Are such magnetic field strengths plausible? If the fields of the magnetic A stars are fossil (which unfortunately is still unclear), sufficiently strong fields might also exist in the cores of solar type stars. Even if the initial fields (on the ZAMS) are lower than these values, however, differential rotation will increase the field strength quickly to values that have an effect on rotation. Whether initially present in the star or developing later by core contraction, differential rotation winds up the field lines, increasing the field strength. This problem has been studied in various forms since the '50s. Winding up of an axially symmetric poloidal field into a predominantly azimuthal field by differential rotation produces an opposing torque that is linear in the number of differential turns made, as in a harmonic oscillator. The result is an oscillation of alternate winding up and unwinding at a period given by the Alfvén travel time through the star (Mestel, 1953), where the Alfvén speed is that of the poloidal field (which is unaffected by the winding-up). Since Alfvén waves travel decoupled from each other, each on its own magnetic surface, the oscillation period is different on each magnetic surface. The oscillations on these surfaces therefore get out of phase after a few oscillations, and the length scale across the surfaces decrease as . In a finite time, dissipative processes across the surfaces become important, and the oscillation damps out by phase mixing (Spruit 1987, Charbonneau & MacGregor 1993, Sakurai et al. 1995). The net effect of the process is that the component of differential rotation along the field lines is damped out, on a finite time scale, and this can happen with an initial field that is much weaker than estimate (3).

3.4. Magnetic shear instability

Another possibility is that a turbulent field is generated by the same magnetically mediated shear instability that has been shown to operate effectively in accretion disks (Hawley et al. 1995, Matsumoto & Tajima 1995, Brandenburg et al. 1995). The conditions for magnetic shear instability to exist in a star have already been studied in detail by Acheson (1978, 1979) though the proper interpretation of this instability (Balbus & Hawley, 1992) was not clear at the time (see, however, Fricke, 1969). In the context of stellar interiors, it has been studied again recently by Kato (1992), Balbus & Hawley (1994) and Urpin (1996). Wherever this instability exists it will lead to very rapid growth (on the differential rotation time scale) of a turbulent magnetic field, which then acts on the differential rotation like an effective viscosity.

Acheson's (1978) analysis of the instability conditions includes (unlike the more recent works) the effects of thermal and magnetic diffusion and of viscosity. The inclusion of thermal diffusion is especially important since it makes the instability appear under much wider conditions. This is seen from Acheson's condition (7.27, a special case of his more general condition), which is equivalent to

where

N is the buoyancy frequency, and the magnetic and thermal diffusivities, and H the pressure scale height. This condition holds for low viscosity (), for an azimuthal field at the equator of the star. The first term on the left hand side represents the magnetic shear instability, the second term Parker instability (magnetic buoyancy instability). For weak fields, this second term is negligible. The right hand side shows the stabilizing effect of the stratification, which, however, is partially undone by thermal diffusion (for adiabatic perturbations, the factor would be replaced by unity). Since photons diffuse so much more effective than the magnetic field, the instability is present much more widely than in an adiabatic treatment. The instability, however, is able to grow only on length scales sufficiently small that thermal diffusion is important. This somewhat limits its effectiveness, and it may be that the effective viscosity it produces is not much larger than the viscosity produced by hydrodynamic shear instabilities (Zahn, 1974) under the same conditions. These questions could, in principle, be readily addressed by an appropriate numerical simulation.

Because magnetic fields are so effective at transmitting torques, already at low field strengths, differential rotation can survive over a large number of rotations only in regions where the radial field component is very small. In order to allow the core in a giant to rotate substantially faster than its envelope, one must find a reason why it could have been so accurately `shielded' magnetically, over the entire life of the star on the giant branch.

While the arguments given here do not constitute a proof, I feel they are sufficiently strong that approximately uniform rotation (modulo a factor of a few) is a reasonable hypothesis, compared with the traditional assumption in which the core of a giant rotates - times faster than its envelope.

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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