that rapidly rotating stars should be well mixed by this circulation, and that

they would be prevented from becoming red giants. But soon Mestel (1953)

showed that the inhomogeneities produced through the burning of hydrogen

in the core would interfere with the circulation and could eventually stop it

(see also Mestel 2003).

A similar circulation may also be induced at the boundary of a radiation

zone, by a di¬erentially rotating convection zone. This is observed in the

Sun, where acoustic sounding has shown that the rotation rate changes from

latitude-dependent in the convection zone, to almost uniform in the radiative

interior below, with the transition occurring in a thin boundary layer, the

tachocline. In the absence of other stresses, such as due to a magnetic ¬eld

or to waves, the circulation in the tachocline has an overturn time given by

(Spiegel & Zahn 1992)

’1 4

r2 N 2 ∆„¦ ∆r

ttac = , (14.3)

K „¦2 „¦ r

with ∆r being the thickness of the tachocline, ∆„¦ the applied di¬erential

rotation, K the thermal conductivity, and N the buoyancy frequency, which

in a chemically homogeneous region is given by

gδ

(∇ad ’ ∇).

N2 = (14.4)

HP

We have used the classical notations: g for the gravity, HP for the pressure

scale-height, ∇ = d ln T /d ln P for the gradient of temperature T with re-

spect to pressure P ; also ∇ad = (‚ ln T /‚ ln P )ad , the derivative being for

an adiabatic change, and δ = ’(‚ ln ρ/‚ ln T )P , where ρ is density.

14.2.3 Turbulence caused by di¬erential rotation

In general, the rotation regime achieved by the interplay of the circulation

discussed above and the applied torques is not uniform, and the shear of

di¬erential rotation is liable to various instabilities, which will generate tur-

bulence and therefore mixing. For a complete review of these instabilities,

we refer the reader to Pinsonneault (1997). Here we shall consider only

those which apparently play a major role, namely the shear instabilities.

14.2.3.1 Turbulence produced by the vertical shear

Let us ¬rst consider the instability produced by the vertical shear, „¦(r).

This instability is very likely to occur, because the Reynolds number char-

Mixing in stellar radiation zones 209

acterizing such shearing ¬‚ows in stars is very high, due to the large sizes

involved. Depending on the velocity pro¬le, the instability may be linear,

or of ¬nite amplitude. In the absence of strati¬cation, turbulence can be

sustained whenever the Reynolds number Re = wl/ν is larger than about

Rec = 40, as it has been discussed by Schatzman, Zahn & Morel (2000).

(The Reynolds number Re is expressed here in terms of the velocity w and

the size l of the largest turbulent eddies, and ν is the kinematic viscos-

ity.) However a radiation zone is stably strati¬ed, and that strati¬cation

acts to hinder the shear instability. In an ideal ¬‚uid, i.e., without thermal

dissipation, the instability occurs only if

N2

¤ Ric , (14.5)

(dVh /dz)2

where Vh is the horizontal velocity, z the vertical coordinate, and N was

introduced in equation (14.4). This condition is known as the Richardson

criterion, and the critical Richardson number Ric is of order unity.

In a stellar radiation zone, this criterion is modi¬ed because the perturba-

tions are no longer adiabatic, due to radiative leakage. When the radiative

di¬usivity K exceeds the turbulent di¬usivity Dv = wl, the instability cri-

terion takes the form (Townsend 1958; Dudis 1974; Zahn 1974; Ligni`res ete

al. 1999)

N2 wl

¤ Ric . (14.6)

(dVh /dz)2 K

From the largest eddies which ful¬ll this condition, one can deduce an es-

timate for the turbulent di¬usivity acting in the vertical direction in the

radiation zone of a star:

2

K d„¦

Dv = wl = Ric 2 sin2 θ , (14.7)

N d ln r

provided that Dv ≥ Rec ν = 40ν.

The instability criterion (14.6) holds in regions of uniform composition,

where the stability is enforced only by the temperature gradient; when the

molecular weight µ varies with depth, it seems at ¬rst sight that one should

replace this criterion by the original one, equation (14.5), where now the

buoyancy frequency is dominated by the gradient of molecular weight:

g• d ln µ

N 2 ≈ Nµ =

2

,

HP d ln P

with • = (‚ ln ρ/‚ ln µ)P,T . As Meynet & Maeder (1997) pointed out,

this severe condition would prevent any mixing in early-type main-sequence

Zahn

210

stars, contrary to what is observed. We shall see below how the stabilizing

e¬ect of µ-gradients can be weakened too.

14.2.3.2 Turbulence produced by the horizontal shear

Likewise, the horizontal shear „¦(θ) will also generate turbulence, most likely

through a ¬nite-amplitude instability, because most plausible rotation pro-

¬les are linearly stable. What type of turbulence then will occur is still a

matter of debate. Its vertical component will be constrained through the

strati¬cation, according to condition (14.6), and therefore it is likely that

this turbulence will be anisotropic, with much stronger transport in the

horizontal than in the vertical direction. In other words, the turbulent dif-

fusivity is a tensor, with its horizontal component Dh much larger than the

vertical one Dv .

Such anisotropic turbulence will interfere with the advective transport due

to the meridional circulation, and it will turn it into a di¬usion, as it was

shown by Chaboyer & Zahn (1992). Assuming that the vertical velocity of

the circulation is given by ur (r, θ) = U (r)P2 (cos θ), the resulting di¬usivity

will be

1 (rU )2

De¬ = , (14.8)

30 Dh

provided that rU ≥ Dh . Unfortunately, a reliable prescription for the hori-

zontal di¬usivity Dh is still lacking, and when modeling this e¬ect one has

to make use of a free parameter.

Another property of such anisotropic turbulence is that, by smoothing

out chemical inhomogeneities on level surfaces, it reduces the stabilizing

e¬ect of the vertical µ-gradient. The Richardson criterion for vertical shear

instability then takes the form (Talon & Zahn 1997)

2

wl wl 2 dVh

N ¤ Ric

2

NT + , (14.9)

Dh µ

K + Dh dz

2

where NT stands for the thermal part of the buoyancy frequency given by

equation (14.4). From this criterion one deduces, as before, the vertical

component of the turbulent viscosity:

’1 2

2

2 Nµ

NT d„¦

2

Dv = Ric + sin θ . (14.10)

K + Dh Dh d ln r

A similar prescription has been proposed by Maeder (1997).

Mixing in stellar radiation zones 211

14.3 Rotational mixing

The three causes of mixing which have just been discussed are intimately

linked with the (di¬erential) rotation. Therefore, when modeling the evolu-

tion of a star including these mixing processes, it is necessary to calculate

also the evolution of its rotation rate „¦(r, θ). The latter changes with time

because angular momentum is advected by the large-scale circulation. At

¬rst sight, the problem looks very simple to handle, just dealing with lami-

nar ¬‚ows. However, as already mentioned, the di¬erential rotation which is

generated by this advection generates turbulence, which in turn transports