e

tation law satis¬ed the Rayleigh criterion d(„¦ 2 )/d > 0, then rotation

would act to inhibit the axially symmetric modes that would otherwise grow

in a superadiabatic region. Cowling derived for a uniformly rotating system

the modi¬cation to the Schwarzschild instability criterion for local distur-

bances of the form exp[i(l + m φ + nz)], stressing that for m = 0, the

constraint of detailed angular momentum conservation is relaxed by the ef-

fect of azimuthal pressure gradients. His analysis shows that motions with

m = 0 and n 0 “ whirling motions in tubes parallel to the rotation axis

“ are hardly a¬ected by the rotation. This local linear analysis demon-

strates both inhibition and persistence of instability, but cannot predict the

expected latitude-dependent reduction in the e¬ciency of heat transport,

with the consequent departure from strict barotropy.

Several workers (Weiss, Durney & Roxburgh, Moss & Vilhu) had empha-

sized that quite modest departures from adiabaticity could be signi¬cant,

especially for variation with latitude. More recently, anisotropic turbulent

heat conductivity (Brandenburg et al. 1992a) is incorporated into a com-

prehensive study by G. R¨diger & L.L. Kitchatinov (Ribes 1994, p. 27).

u

Like the earlier workers, they begin with the toroidal dynamics, ignoring

laminar circulation, computing the coe¬cients in QΛ , QΛ from a model of

rφ θφ

inhomogeneous turbulence, and solving (6.37) for a convective shell subject

to stress-free boundary conditions. For their chosen parameters, the result-

ing „¦-contours are close neither to the Biermann nor the Taylor-Proudman

forms, but are approximately radial. For consistency, the circulation v p

driven by the non-irrotational centrifugal ¬eld must be slow enough to jus-

tify neglect of the laminar advection of angular momentum in (6.36). The

curl of the poloidal equation of motion is

D(v p ) = (1/ρ2 )(∇ρ — ∇p) + (‚„¦2 /‚z)t, (6.38)

where the term on the left is the curl of the frictional drag on vp and so can

be written in terms of Qν . After non-dimensionalization, the term in „¦2 has

ij

as coe¬cient the Taylor number Ta = (2„¦R2 /νt )2 . For a moderate rotator

like the Sun, with the circulation slow, the two terms on the right of (6.38)

Mestel

96

must be in approximate balance: the observed equatorial acceleration then

requires the poles to be somewhat hotter than the equator, as observed. The

actual temperature distribution is determined by the energy equation (with

viscous heating ignored):

∇ · (F conv + F rad ) + cp ρvp · ∇(∆T ) = 0, (6.39)

where ∆T is the superadiabaticity, F rad the usual radiative heat ¬‚ux, and

Ficonv = ’cp ρχij ‚(∆T )/‚xj . (6.40)

The anisotropic thermal conductivity tensor χij has to be constructed from

a model of the in¬‚uence of rotation on the turbulent convection (Kitchatinov

et al. 1994). The simultaneous solution of all the equations then does yield

a hotter pole, and con¬rms that for Ta < 107 , the slow-circulation model “

yielding equatorial acceleration “ is indeed a good approximation; but for

larger Ta, even though the domain is not strictly adiabatic, the rotation

approximates to the Taylor-Proudman law „¦ = „¦( ), and now with „¦ < 0

“ equatorial deceleration. General agreement with the solar data is shown

when „¦— 2.3, as compared with the actual estimated solar value of 6.

The most recent studies of fully three-dimensional compressible convec-

tion, do not in fact yield behaviour reminiscent of the phenomenological

mixing-length picture, yet it appears that the classical mixing-length for-

mulae remain reliable as a convenient ex post facto parametrization for esti-

mating velocities and ¬‚uxes. A more sophisticated terminology may emerge

from further studies of the solar rotation and any associated meridional mo-

tions, crucial for solar dynamo theory, and for the appropriate extensions to

the younger, more rapidly rotating late-type stars. The predicted changes

in „¦( , z) at high „¦— should already make one cautious about too cavalier

an extrapolation of results for the solar dynamo to more rapid rotators.

6.6 The solar tachocline

Let me now refer brie¬‚y to the transition from the latitude-dependent ro-

tation of the solar convective zone to the nearly uniform rotation inferred

for the bulk of the radiative core. The various studies in the literature

illustrate the di¬erent possible approaches to stellar rotation, as outlined

above. When studying the solar core, most workers are prepared to take

the rotation at the base of the zone as prescribed. Spiegel & Zahn (1992)

follow Spiegel (in Perek 1968, p. 261) in presenting ¬rst what I would call

a non-magnetic, inviscid, Eddington-Sweet approach to the history of the

Stellar rotation: a historical survey 97

inner solar rotation. After a transient phase, the laminar meridional veloc-

ity of the gas is related to the local angular velocity ¬eld by an equation

derived from the quasi-steady (6.14) with = 0, and with the dominant

contribution to v coming from the term in „¦ (cf. the end of Section 6.1);

while the evolution of the „¦-¬eld is given by (6.30). It is found that af-

ter a solar lifetime, during which the convective envelope has been subject

to the magnetic solar wind torque, the tachocline would have increased in

thickness to 2 ’ 3 — 105 km, far greater than the observationally unresolved

thickness. In the second part, the authors introduce a strong turbulent fric-

tion into (6.30). An isotropic friction would only increase the spreading

rate of the tachocline, but the authors follow Zahn™s earlier argument and

take the horizontal component to be far greater than the vertical. They

predict a satisfactorily thin tachocline and an internal angular velocity in-

termediate between the polar and equatorial values in the convection zone,

in agreement with the interpretation of helioseismic observations. But as

seen, the modelling of shear turbulence as acting to produce „¦ horizontally

uniform is not universally accepted, e.g. by Gough & McIntyre (1998), who

when considering the same problem, ¬nd themselves entitling their paper:

˜Inevitability of a magnetic ¬eld in the Sun™s radiative interior™.

Helioseismology and now also asteroseismology are providing more and more

stringent tests of stellar astrophysics. Let me conclude by noting with plea-

sure how much Douglas has contributed to the exploitation of this gold-mine.

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