in meridian planes “ vertical and north-south “ and also the azimuthal, east-

west component; hence ˜U small™ implies a small non-uniform rotation as

viewed in the inertial frame.

The geophysicist also ¬nds it convenient to take curl of (6.19), which yields

the rate of change following the motion u of the relative vorticity ω ≡ ∇—u:

dω/dt = (ωa · ∇)u ’ ωa (∇ · u) + (∇ρ — ∇p)/ρ2 + ∇ — F/ρ, (6.20)

where

ωa = ω + 2„¦c (6.21)

is the absolute vorticity, measured in the inertial frame. In the small Rossby

number limit, ωa 2„¦c , and in a friction-free steady state, (6.20) reduces

to

(2„¦c · ∇)u ’ 2„¦c ∇ · u + (∇ρ — ∇p)/ρ2 = 0. (6.22)

In the ¬‚uid dynamicist™s language, in a steady state, the generation of rel-

ative vorticity by baroclinicity is equal and opposite to a similar generation

by the relative motions u, through stretching and twisting of the already

existing vorticity 2„¦c .

As a special case, suppose that the system were symmetric about the axis;

then ∇ρ — ∇p is in the azimuthal (φ) direction, with (6.22) ¬xing ‚uφ /‚z;

and if the baroclinicity is zero, then uφ is a function just of . In the

inertial frame, this is equivalent to a total local angular velocity „¦( ) =

„¦c + uφ ( )/ . Thus we recover (for the case of small shear) the results

leading up to the relations (6.4): the existence of a centrifugal potential

implies barotropy, and vice versa, a special case of the Taylor-Proudman

Mestel

82

theorem (Pedlosky 1982). In general, the vorticity equation (6.22) ¬xes the

z-variation of the velocity components in terms of the baroclinicity. With

the same approximations, the original momentum equation (6.19) yields

the ˜geostrophic approximation™, in which e¬ective hydrostatic equilibrium

under the centrifugally modi¬ed gravity holds in the vertical direction n,

and the meridional and azimuthal components of the horizontal velocity uH

satisfy

uH = (1/f ρ)(n — ∇p), (6.23)

where f is the local Coriolis parameter 2„¦c cos θ. An inertial frame ob-

server would describe one component of (6.23) as balance of the meridional

horizontal pressure gradient against the horizontal component of the extra

centrifugal acceleration due to the small uφ , and the other as showing how

in a steady state, the e¬ect of an azimuthal pressure gradient must be o¬set

by the advection of angular momentum.

The point to emphasize is that in the geophysical approach, the variations

in temperature and density are regarded as the cause of the whole low-

Rossby number motion u, which is therefore given the term the ˜thermal

wind™. This is entirely appropriate in a medium of low optical depth, since

the processes directly a¬ecting the temperature “ non-uniform solar heating,

cloud formation “ are only weakly dependent on the motions; rather, it is

the motions which respond to the consequent changes in the pressure-density

¬eld. If a steady state has been reached, then it is the momentum equation

which ¬xes both the deviation from uniform rotation and also the nearly

horizontal meridional velocity. By contrast, in a stellar radiative zone, it

is the centrifugal ¬eld which is the cause of the deviation from spherical

symmetry in the (p, ρ) ¬eld, and so also of the ¬‚ow of heat leading to the

small pressure variations that drive the meridian circulation. The circulation

is present in an axisymmetric star “ it does not arise from the the need to

balance an azimuthal thermal pressure gradient by advection of angular

momentum; but equally, this advection must be included when studying the

angular velocity ¬eld as a function of position and time (Section 6.4).

6.3 Steady circulation and the mixing problem

Returning to the stellar problem, in the late 40s and 50s interest in the cir-

culation problem was stimulated by studies of stellar evolution. It was often

assumed, explicitly or tacitly, that an isolated, rotating, early-type main

sequence star would remain chemically homogeneous, in spite of the strong

temperature dependence of the rate of energy liberation. The short turn-

Stellar rotation: a historical survey 83

over time of turbulent convective motions was clearly a su¬cient argument

for e¬ective homogeneity of the core; and from a rather super¬cial reading of

Eddington™s 1929 paper, it was supposed that the circulation currents would

carry nuclear processed material from the core into the envelope so ensuring

that the gradient of mean molecular weight µ remained small throughout

the whole star. However, a homogeneous star with steadily increasing µ

follows a path in the Hertzsprung-Russell diagram that is up and to the left

of the zero-age main sequence. Such models could be appropriate for the

comparatively few ˜blue stragglers™, but not for the far more numerous red

¨

giants. It was in fact already clear from pioneering work by Opik, and later

from the studies by Hoyle & Lyttleton, Li Hen & Schwarzschild and Bondi

& Bondi, that a non-homogeneous µ-distribution was in principle able to

account for the extended envelopes and the associated low surface tempera-

tures of red giant stars. These models consisted essentially of a helium-rich,

nearly homogeneous inner part, surrounded by a hydrogen-rich envelope.

Such a structure could arise if a highly evolved but homogeneous star “

with a convective core, still generating energy by the C-N cycle, and a sur-

rounding radiative envelope “ subsequently acquired a hydrogen envelope,

e.g. via gravitational accretion of interstellar matter. However, to form a gi-

ant with a large radius, the mass accreted during the nuclear lifetime would

have to be improbably high “ of the same order as that of the accreting star.

More plausibly, a giant structure could arise if for some parameter range, ro-

tational mixing was limited to roughly half the envelope. New light was shed

on the whole area of stellar evolution from photo-electric studies of the H-R

diagrams of evolved globular and galactic clusters, especially by Sandage

and collaborators, which suggested strongly that evolution of stars within a

given cluster is best described to a zero-order approximation as depending on

just one parameter (the stellar mass). Attempts to explain a universal phe-

nomenon like the giant sequence by appealing to an adventitious factor such

as mass accretion ceased to carry conviction, just on observational grounds.

Following the papers by Sandage & Schwarzschild (1952), Tayler (1954),

and especially by Hoyle & Schwarzschild (1955), opinion swung strongly to

the view that normal stellar evolution occurs with negligible non-turbulent

mixing.

¨

The work of Sweet (1950) and Opik (1951) brought out clearly that the

characteristic time of the Eddington-Vogt currents in a uniformly rotating

star is of order „KH /µ, where „KH is the global Kelvin-Helmholtz time and

µ an average centrifugal parameter. It was also early recognized that the

theory needed to be completed by consideration of the azimuthal component

of the equation of motion. As the ¬rst task is to ¬nd an upper limit to the

Mestel

84

e¬cacy of ˜rotational mixing™ by the laminar E-V-S circulation, we begin by

assuming that a steady circulation is maintained by a torque, due e.g. to a

weak magnetic ¬eld, able to o¬set the advection of angular momentum by

the circulation itself and so keep „¦ nearly uniform.

Prima facie, a rough criterion for rotational mixing to be important is

„KH /„nucl ≡ ·,

µ (6.24)

where „nucl is the characteristic time for the nuclear evolution of a star, kept

homogeneous by mixing currents. Since · 10’3 , it appeared that whereas

slow or moderate rotators would su¬er virtually no mixing, a rapid rotator

“ e.g. an A-type star with a rotation period P of 1 day “ would remain

homogeneous, and a star with say P 2 days would acquire an inner nearly

homogeneous zone of mass well above that of the turbulent convective core.

This would imply that the dispersion of rotation periods among the more

rapid rotators could yield a noticeable spread in the evolutionary tracks.

However, a crucial new feature is the back reaction of the distribution of µ

“ brought about by the combined e¬ect of nuclear processing in the core and

advection by the circulation “ on the star™s thermal ¬eld and so on the buoy-

ancy forces driving the circulation (Mestel 1953, Mestel & Moss 1986). In a

non-rotating star, hydrostatic equilibrium requires that T /µ be spherically

symmetric, so that a non-spherical µ-distribution will imply a corresponding

θ-dependence in T . As in the theory of Section 6.1, the radiative heat ¬‚ux

F now has both vertical and horizontal components; the divergence ∇ · F in

general varies horizontally and does not vanish, so yielding a ˜µ-current™ ve-

locity v µ , analogous to the E-V-S velocity v „¦ . The instantaneous µ-current

¬eld is a complicated functional of the µ-distribution and its gradients; in

particular, its sign at points with the same θ can change from negative to

positive as r changes. One simple result emerges at once. If the non-spherical

µ-distribution is due just to the distortion by a quadrupolar P2 (cos θ) per-

µ

turbation of an existing µ0 (r)-¬eld with µ0 (r) < 0, then the sign of vr is

such as to oppose the distortion and so to restore spherical symmetry in µ:

a µ0 (r) ¬eld with a negative gradient is in this sense ˜secularly stable™.

In a rotating star, one may proceed by superposing the e¬ects of the

„¦- and µ-perturbations, yielding a velocity ¬eld v depending on both the

instantaneous „¦- and µ-¬elds. With all the nuclear processing taking place in

the convective core, then in the radiative envelope, the µ-derivative following