a stellar lifetime. In the asymptotic steady state, even the limited freedom

of isorotation rather than uniform rotation is in fact unlikely to persist. In

an axisymmetric state, magnetic ¬eld lines which penetrate slightly into a

convective zone can interchange angular momentum through coupling with

the turbulence; and even a modest departure from axisymmetry will tend

to enforce near uniform rotation through the action of toroidal magnetic

pressure gradients (Moss et al. 1990).

However, again a large departure from axisymmetry yields qualitatively

new results. Whereas in the axisymmetric problem, rotational shearing has

only a weak e¬ect on the basic poloidal ¬eld, rotational distortion of a highly

oblique ¬eld can lead to juxtaposition of oppositely directed lines of the basic

¬eld, leading to accelerated Ohmic decay (K.-H. R¨dler, in Guyenne 1986, p.

a

569). A weakish ¬eld that is initially highly oblique could thus be converted

into a more nearly aligned ¬eld, simply through the accelerated decay of the

perpendicular component; whereas a stronger ¬eld will be able to reverse the

initially imposed non-uniform rotation before the accelerated Ohmic decay

gets under way (Moss et al. 1990)

The postulated presence of some magnetic ¬‚ux across the whole H-R

diagram is by no means unreasonable (e.g. Mestel 2002). In particular,

braking of the rotation of late-type stars, either as a consequence of magnetic

activity (Schatzman 1962) or through magnetic coupling with a wind (Mestel

1967, 1999; Weber & Davis 1967; Mestel & Spruit 1987) is accepted as a basic

phenomenon by all workers. Likewise, the ability of even a weak magnetic

Stellar rotation: a historical survey 89

¬eld to transport angular momentum e¬ciently through a stellar radiative

zone is recognized (e.g. Mestel & Weiss 1987; Spruit, in Durney & So¬a

1987; Spruit 1999). Certainly, this does not mean that studies outlined in

Section 6.4.2 below are redundant, but rather that it should be stressed that

neglect of magnetic e¬ects does put severe upper limits on the strength of

any ¬eld present.

In late-type stars, dynamo action in both the convective zone and in the

tachocline, at least in part analogous to that observed and inferred in our

Sun, will generate and maintain time-dependent magnetic ¬elds. Such ¬elds

will have the topology with poloidal-toroidal ¬‚ux linkage that appears to be

necessary though not su¬cient for stability. It is not clear whether the ¬elds

of the strongly magnetic early-type stars are slowly decaying ˜fossils™, or are

themselves being maintained by contemporary dynamo action. A fossil ¬eld

may also be present e.g. in the radiative cores of the Sun and other late-type

stars. Magnetic ¬elds are notoriously subject to instabilities, and the con-

siderable literature (e.g. Wright 1973; Acheson 1978; Tayler 1980; Spruit

1999; Mestel 1999 and references therein) has not given an unambiguous

answer as to whether non-dynamo maintained ¬elds can persist over stellar

lifetimes. Spruit (1999) appeals to observation, arguing that the magnetic

white dwarfs are ˜fairly strong evidence™ that long-term stable ¬elds do exist

in main sequence stars.

6.4.2 Non-magnetic radiative zones

Consider now the problems of a rotating radiative zone, with magnetic ef-

fects assumed ignorable. The basic dynamical restrictions on the allowed

distribution of angular momentum are the Solberg-Høiland criteria (e.g.

Tassoul 2000). They are essentially a generalization to a rotating star of the

Schwarzschild criterion for stability against convection, restricting the direc-

tion of the gradient of the speci¬c angular momentum h = „¦ 2 to a domain

bounded by the direction of e¬ective gravity and the outwardly pointing di-

rection of the gradient of speci¬c entropy. They are derived by considering

adiabatic perturbations, for which buoyancy forces act to oppose instability.

However, as discussed originally by Goldreich & Schubert and by Fricke,

and later by Smith & Fricke, on a thermal time-scale a displaced element

exchanges heat with its surroundings and so loses its buoyancy. Although,

as seen, it is possible in general to ¬nd hydrostatic equilibria with arbitrary

rotation ¬elds by having appropriate temperature and density variations, it

Mestel

90

is found that only those with

‚h2 /‚ > 0, ‚h/‚z = 0 (6.29)

are secularly stable (e.g. Tassoul 2000). In an incompressible ¬‚uid, the

¬rst of these is the Rayleigh stability criterion, while the second is a part of

the Taylor-Proudman condition for equilibrium (cf. Section 6.2). In a stably

strati¬ed gas, there exist motions that are su¬ciently slow for heat exchange

to annul the stabilizing buoyancy forces. As summarized by Fricke, one can

¬nd which states are secularly stable by solving the dynamical problem in

the corresponding incompressible system. (Fricke studies also axisymmetric

rotating magnetic stars, satisfying the law of isorotation, ¬nding that only

the special case with „¦ the same constant on each ¬eld line is secularly

stable.)

Suppose that „¦ is initially uniform and large enough for the E-V-S cir-

culation time-scale to be well below a stellar lifetime. If viscous e¬ects are

negligible, the advection of angular momentum changes the „¦-¬eld, accord-

ing to

‚„¦/‚t = ’v · ∇(„¦ 2 2

)/ . (6.30)

It was argued by Busse that the rotation ¬eld would evolve towards a

circulation-free state (cf. Section 6.1); but these states are themselves secu-

larly unstable according to the criteria (6.29). Again, Osaki and later Kip-

penhahn & Thomas (in Lamb et al. 1981) argued that in the low-density

surface regions of an early-type star, a small deviation from uniform rota-

tion is su¬cient to cancel to second order the terms ∝ ρ/ρ, leaving just a

¯

velocity of the Sweet order; but again this rotation law is secularly unstable.

The G-S-F thermally-driven instabilities are the seed of a form of weak

turbulence, which also acts to redistribute angular momentum, in a time that

was early estimated to be at least of the order of the Kelvin-Helmholtz time.

James & Kahn (1970, 1971) increased this to the more plausible estimate

of the standard E-V-S time, a result supported later by Kippenhahn et al.

(1980) and by Kippenhahn & Thomas. Thus it appeared that one should

add to (6.30) a quasi-viscous term with a characteristic time of the same

order as the circulation time.

Besides the G-S-F instabilities, which will occur under axisymmetric dis-

turbances, there are the possible non-axisymmetric shear instabilities, in

which the energy in the di¬erential rotation velocity vφ = „¦ is used to

overcome the stabilizing buoyancy forces, and to overturn a stably strati¬ed

ρ(r)-distribution. There is as yet no consensus as to their likely importance.

The standard criterion allowing this Kelvin-Helmholtz-type instability is

Stellar rotation: a historical survey 91

that the Richardson number

JR = [g(dρ/dr ’ (dρ/dr)ad )]/ρ(dvφ /dr)2 (6.31)

be less than some critical value, usually taken as 1/4 (e.g. Chandrasekhar

1961). This can be written equivalently cos θ(‚ log „¦/‚ log r) > 2N/„¦ where

N is the Brunt-V¨is¨l¨ frequency. By allowing for heat di¬usion, Zahn

a aa

(1975) found a much reduced upper limit. Jones and Acheson noted that

these results tacitly assume ¬nite amplitude disturbances, whereas for in-

¬nitesimal disturbances, a much stronger vertical shear can be tolerated.

But assuming shear instability does set in and develop, what rotation law is

achieved? Noting that the buoyancy stabilizes only in the vertical direction,

Zahn (1975) argued that the turbulence is likely to smooth out the horizon-

tal shear, yielding a rotation law „¦(r). Following to some extent an earlier

treatment by Sakurai, Zahn (1992) then studied the consequent evolution

of „¦(r, t) within a star, the competing processes being magnetic braking by

a stellar wind, shear turbulence, and advection of angular momentum by

the instantaneous E-V-S circulation, constructed essentially as outlined in

Section 6.2, with the µ-current e¬ect included.

However, the assumption „¦(r) is disputed. Kippenhahn & Thomas cited

experiments on rotating liquids as showing that variations of „¦ along equipo-

tentials are not spontaneously wiped out, but could retain values that de-

pend on the history of the star. By contrast, Tassoul (2000) bases many

studies in this area on an appeal to the analogue of the geophysicist™s baro-

clinic instability, to which the „¦(r) ¬eld is subject for almost all positive

values of the Richardson number (6.31). And in a recent paper on the

solar tachocline, Gough & McIntyre (1998) appeal to both ˜strong theoreti-

cal arguments™ and a ˜wealth of observations™ of the terrestrial stratosphere

showing that two-dimensional turbulence drives the angular momentum dis-

tribution away from rather than towards uniform rotation.

It is both proper and desirable that there be these parallel studies of

stellar rotation, with magnetic e¬ects completely ignored; but it should

simultaneously be stressed that at least for radiative zones, there is implicit

a severe upper limit on the strength of any ¬elds actually present. Likewise,

it is incumbent on those who stress the importance of magnetic e¬ects in

stellar radiative zones also to keep a watchful eye on potential hydromagnetic

instabilities (e.g. Spruit 1999).

A principal aim of these studies is to elucidate any possible e¬ect of rota-

tion on stellar evolution, in particular supplying theoretical background to

well-known problems such as the Li abundance in main-sequence stars, and