Mestel

92

Zahn 1992 and in Durney & So¬a 1987; and many other papers).

6.5 Rotating convective zones

The canonical description of turbulent convection is in terms of the mixing-

length l (the macroscopic analogue of the mean-free-path » in kinetic the-

ory) the associated turbulent velocity vt , and a corresponding convective

˜turnover time™ „c = l/vt . The analogy itself suggests the introduction of

a kinematic ˜eddy-viscosity™ νt lvt /3. Models of the solar convection

3 — 103 cm s’1 , whence

109 cm, vt

zone yield as typical values l

1012 cm2 s’1 , as compared with a microviscosity (kinetic or radiative)

νt

103 . If the analogy were perfect, the eddy viscosity would smooth out

shear over a scale L in the time L2 /νt : acting alone, it would enforce uni-

form rotation of a convective zone in a time much below a stellar lifetime.

Biermann (1951) gave a simple argument to illustrate how this is brought

about in a gas with isotropic random velocities. In a steady state, equal

numbers of particles per second cross unit area of a ¬xed spherical surface S

in each direction, and likewise they must carry the same amount of angular

momentum. Clearly, those particles which actually arrive at the surface

from outside must have in the mean less angular momentum than that

possessed by the particles in the place of origin. The reason becomes clear if

for convenience we use a frame rotating with the assumed uniform angular

velocity „¦c of the star, and for simplicity study the motion of particles near

the equatorial plane in between two collisions, separated by the mean-free-

path ». Over the distance »/2, an incoming particle with velocity vm has

its component perpendicular to the rotation axis rotated by the Coriolis

acceleration at the rate ’2„¦c , acting for the time »/2vm . It follows that a

particle arriving at S at right angles must have started moving after its last

collision at an angle „¦c »/vm to the radius vector and in the sense opposite

to the rotational velocity. The same result holds for all directions to S. This

ensures that the particles which actually cross S have on average exactly the

same angular momentum as the particles on S, as clearly must be the case

in a stationary state.

The crucial point (as noted earlier by Wasiutynski and emphasized by

Biermann), is that the argument assumes isotropy of the particle velocities in

the rotating frame. If instead, for example, after each collision, the direction

parallel to the cylindrical radius vector were preferred, then the same

Coriolis acceleration would give particles crossing S the angular momentum

(in the inertial frame) „¦c 2 + („¦c »/vm )vm „¦c ( + »/2)2 , just the

Stellar rotation: a historical survey 93

mean angular momentum in a uniformly rotating star of particles at distance

( + »/2). There would then be an in¬‚ow of angular momentum, clearly

inconsistent with the assumption of a steady state in uniform rotation.

However, as was ¬rst pointed out by by Lebedinski and by Wasiutynski,

the turbulence arising from instability under a gravitational ¬eld would in-

deed be anisotropic, with l and vt having di¬erent values in vertical and

horizontal directions, and so should lead to a non-uniform rotation ¬eld.

Generalizing from the treatment of the ¬‚ow in the equatorial plane, Bier-

mann wrote the turbulent ¬‚ux of angular momentum as the sum of an

isotropic part ’A1 2 ∇„¦ and a monotropic part ’A2 g e (ˆ e ·∇( 2 „¦)), where

ˆg

ˆ

ge is the unit vector in the direction of e¬ective gravity. In a steady state

(determined by just the action of the eddy viscosity) the two terms must

sum to zero, yielding ∇„¦ parallel to g e ; and in a slow rotator like the Sun,

ˆ

ˆ

ge is nearly radial, so

„¦ ∝ r ’2A2 /(A1 +A2 ) . (6.32)

The treatment has been generalized by R¨ diger (1989) and R¨diger &

u u

Kitchatinov (in Ribes 1994), following the methods introduced by Osborne

Reynolds. The velocity ¬eld in a turbulent domain is written as the sum of

mean and ¬‚uctuating parts:

v = („¦ t + v p ) + v , (6.33)

with the mean velocity having in general a meridional (poloidal) circulatory

component as well as a rotatory (toroidal) part (in which t is a unit vector).

When substituted into the non-linear inertial term, the mean of the terms

in v yields an e¬ective body force ’∇ · (ρQ), where ’ρQij is the Reynolds

stress tensor. The traditional parametrization of Qij is generalized to

Qij = Qν + QΛ . (6.34)

ij ij

Here the di¬usive part Qν depends on the velocity gradients through an eddy

ij

viscosity tensor, and the ˜Λ-e¬ect™ term QΛ = Λijk „¦k is the generalization of

ij

the Lebedinski-Biermann terms that depend on „¦ but not on its derivatives.

In spherical polar coordinates, the terms in Qij , that contribute to angular

momentum transport are

Qrφ = ’νvv r sin θ‚„¦/‚r + QΛ ,

rφ

Qθφ = ’νhh sin θ‚„¦/‚θ + QΛ . (6.35)

θφ

In a slow rotator, QΛ ΛV sin θ„¦ and QΛ 0, with ΛV , νvv and νhh

rφ θφ

all independent of θ. If magnetic torques are ignorable, then in a steady

Mestel

94

state, angular momentum transport by the Reynolds stresses is balanced by

advection by the laminar circulation:

∇ · (ρ„¦ 2

v p + ρ vφ v ) = 0. (6.36)

If the circulation can be ignored, Reynolds stresses alone ¬x the „¦-¬eld by

r ’2 ‚(ρr 3 Qrφ )/‚r + (ρ/ sin2 θ)‚(sin2 θQθφ )/‚θ = 0. (6.37)

In the slow rotation limit, with QΛ 0, the zero-stress condition adopted

θφ

at the surface of a convective envelope and „¦ assumed independent of θ

at the base, (6.37) yields zero stress throughout the zone, with „¦ = „¦(r),

Biermann™s basic result.

Biermann™s treatment was extended by Kippenhahn (1963). While cru-

cial for angular momentum transport, the Reynolds stresses of the sub-

sonic turbulence (the ˜turbulent pressure™) make only a small contribution

to hydrostatic support, which is therefore again given by (6.1) in a weakly

magnetic domain. And if one assumes the zone to be nearly adiabatic,

following the classical Biermann-Cowling analysis for non-rotating systems,

then the surfaces of constant ρ and p coincide and the curl of (6.1) again re-

quires the centrifugal ¬eld to be conservative. Thus the simplest treatment

of the ˜poloidal dynamics™ enforces „¦ constant on cylinders, whereas the

above treatment of the ˜toroidal dynamics™ yielded „¦ constant on spheres.

Kippenhahn argued that the near barotropy of the (p, ρ)-¬eld and the non-

conservative centrifugal ¬eld would together enforce a meridian circulation

with velocities ¬xed by making the drag due to the eddy-viscosity contribute

to the balance of pressure, gravity and centrifugal force. The constructed

vp -¬eld is then fed into (6.36) to yield the deviation of the „¦-¬eld from pure

r-dependence. Equatorial acceleration “ as observed in the Sun “ results

when v p is equatorwards at the surface; this requires „¦ to increase out-

wards, in turn requiring the horizontal mixing-length to be larger than the

vertical.

With realistic solar parameters inserted, this iterative scheme did not

converge. The form used above for the Λ-e¬ect is valid only if the Coriolis

number „¦— = 2„c „¦c 1, whereas the estimated value for the Sun is „¦— > 1.

As „¦— approaches unity there is now a non-zero contribution to QΛ , so that

θφ

the Reynolds stresses, acting alone, will yield a θ-dependent „¦-law, whereas

in the low-„¦— , Biermann-Kippenhahn approach, it is the circulation that

yields departure from the pure r-dependent law (6.32). However, later stud-

ies (e.g. Brandenburg et al. 1990, 1992) showed that as long as adiabaticity

is retained, then even with a more appropriate form for the Λ-e¬ect, the

combined e¬ects of the toroidal and poloidal equations still do not allow the

Stellar rotation: a historical survey 95

isorotation contours to deviate far from Taylor-Proudman cylindrical law.

Thus the pioneering Lebedinski-Biermann-Kippenhahn approach has to be

modi¬ed both for internal self-consistency, and even more in order to tally

with the helioseismological data, which yield an „¦-¬eld consistent with nei-

ther the Taylor-Proudman nor the Biermann laws as adequate zero-order

approximations.

The possible inhibiting e¬ect of rotation on convection had been studied