Hp 8 J 3 dm

where Vnt in deep stellar interiors is generally equal to Vad (see Sect. 5.3). In 1D models, one may consider that the average situation corresponds to P2(cos $) = 0, i.e., cos2 $ = 1/3 so that sin $ = ^J2/3. In general, the specific angular momentum increases outward, thus the oscillation frequency is higher in a rotating medium, consistently with an additional recall force acting on a displaced fluid element. It is to be noted that g is here the gravitational acceleration only, owing to (6.45).

6.4.2 The Rayleigh Criterion and Rayleigh-Taylor Instability

Let us clarify some definitions. The Rayleigh criterion is the criterion, which states the condition for a distribution of angular velocity to be stable (other stabilizing or destabilizing effects being ignored). It is expressed by the condition NQ > 0 (6.50) or by (6.51), the Rayleigh frequency in a rotating medium being the frequency Nq . The Rayleigh-Taylor instability occurs when a denser fluid is supported by a lighter one against a gravitational field or another acceleration. It can occur in very different contexts, for example, when a heavier gas is accreted onto a binary system or when a denser fluid is accelerated in a lighter one, as, for example, in galactic jets or supernova explosions. (Please note that the so-called Rayleigh instability or Plateau-Rayleigh instability concerns something very different, it occurs when surface tension breaks a jet into a stream of droplets.)

We consider the case of a medium of constant density, the sum Nj + N2 is zero and only the term with NQ2 is left in (6.49). The stability condition requires in order that the oscillation motions do not grow exponentially

since sin ô is positive and equal to zero at the pole. If NQ > 0, the lower angular momentum of the upward displaced fluid element brings it back to the equilibrium position. In the opposite case, the higher angular momentum of the displaced element drives it farther. Condition (6.50) means that

- For stability, the specific angular momentum j = —2Q must increase outward.

- One also has from (6.50) for stability, dj ( dQ \ ^ dln Q „

This (or 6.50) expresses the Rayleigh criterion. Q must not decrease too steeply outward, otherwise the angular momentum is higher in the inner layers and the

Rayleigh criterion is violated. If Q is of the form Q ~ r-a, values of a > 2 lead to instability by violation of the Rayleigh criterion. - On the stellar surface, the angular momentum must also increase from pole to equator.

The characteristic timescale of the Rayleigh instability, if present, is the dynamical timescale 1/NQ of the rotating system, i.e., of the order of 1/Q. For the transport coefficient, an expression of the form

may be chosen according to Sect. 10.1.3, where lQ is a characteristic length scale of the rotation motions, e.g., lQ « \Q(dr/dQ)\.

6.4.3 The Solberg-Hoiland Criterion

In the presence of a density stratification, the stability condition is

where & is the colatitude (Fig. 6.4.1). This is the Solberg-Hoiland criterion for con-vective stability. As a matter of fact, it expresses a stability condition with respect to a form of the Rayleigh-Taylor instability. In the absence of rotation, the criterion is just the usual Ledoux criterion (5.26): NJ + Nl > 0. Condition (6.53) means that convective stability is favored by stellar rotation if the outward decrease of Q is moderate according to condition (6.51). In this case, the extent of a convective region is slightly reduced by rotation, with a larger reduction at the equator than in polar regions. Figures 6.5 and 12.8 show the region where convective instability is prevented by rotation. In 1D models, an average value of sin & = \J2/3 may be taken. Criterion (6.53) also implies, if NQ is negative enough, that a region which would have been radiatively stable could be unstable.

Since Nj = 0 implies constant entropy (cf. Sect. 5.1.1), condition (6.53) also implies that on a i constant isentropic surface, the angular velocity increases from pole to equator.

In the region just outside the convective core, both the i and the Q gradients can be very steep and highly dominating in (6.53). This is particularly the case in the advanced stages of nuclear burning. An equilibrium situation between the two gradients may possibly be reached for the following reasons. The high i gradient in itself favors stability, but simultaneously the growth of central density, due to central condensation, leads to an increase of the Q gradient. The steep Q gradient may drive the instability. The resulting fast transport proceeds until stability is restored. Thus, an equilibrium situation is reached with

in the region just outside the core in advanced evolutionary stages. This gives in terms of the / and Q gradients,

The steeper the / gradient, the steeper the Q gradient in such an equilibrium at the limit of the Solberg-Hoiland criterion.

6.4.4 Numerical Simulations

There are two extreme assumptions about rotation in a convective zone:

- Case A: it is generally assumed in 1D stellar evolution models that convective regions are rotating like a solid body. The argument is that the strong turbulent viscosity due to convective motions maintains a uniform distribution of angular velocity.

- Case B: an alternative possibility is that the distribution of the angular momentum is constant in convective regions. The argument is that large-scale motions dominate and conserve their angular momentum, thus establishing a uniform distribution of the specific angular momentum.

These two possibilities are very different. In the first case, one has a constant Q, while in the second one has Q ~ ®-2. In red giants, case B leads to more differential rotation, more shears and thus more mixing at the base of the convective envelope

Between cases A and B, the reality is more complicated as suggested by 2D and 3D numerical simulations of convection, which have been applied to the solar con-vective envelope and to convective cores. Three-dimensional simulations of solar convection by Toomre and Brun [577] show that convection is time dependent with intricate flows, dominated by intermittent plumes of upflows and stronger down-flows extending over much of the shell depth. Time-averaged angular velocity Q distributions show the following features:

- In the equatorial regions, Q(r) increases by 10-15% from radius r/RQ = 0.71 to the surface.

- Away from the equator, Q(r) is lower with a smaller increase from the base of the convective zone to the top.

- In polar regions, Q(r) is constant with depth, being about 30% lower than at the equator in surface.

Turbulent motions are coupled with several cells of meridional circulation in latitude and often two layers of cells in depth. These simulations [577] well reproduce the values Q(r) obtained from helioseismology; however they do not support cases A or B above. One notes that solar rotation is very low and a different picture may apply to fast rotation, as well as to situations where the Mach number is high, like in red giants or supergiants. Numerical Simulations of Convective Cores

Three-dimensional simulations of core convection in 2 M0 stars of different rotation velocities have been made [71] in the anelastic approximation, i.e., for fluid elements in pressure equilibrium. Large-scale motions are dominant, with multicellular structures of meridional circulation, which are orders of magnitude faster than the classical meridional circulation in radiative envelopes. Hydrodynamic simulations [162] indicate faster convective velocities than estimated by the MLT. For moderate and rapid rotation, convective motions are mostly parallel [162] to the rotation axis, rather than vertical. Contrarily to the solar case, small-scale turbulent features are absent and there are no asymmetries between up and downflows. Convective plumes excite [71] gravity waves in the radiative envelope (cf. Fig. 6.3), where slow circulation is also noticed.

The convective cores rotate differentially. Some models [162] suggest that the variations of Q(r) can be represented by Q ~ B-05, with a better agreement for fast than for slow rotation. In the slowly rotating models by Brun et al. [74], a central cylindrical region with slower rotation is present, with significant Q gradients both in radius and in latitude. At the equator, Q(r) increases with radius, while at higher latitude Q(r) is almost flat. At the edge of the convective core, Q at the equator is larger by 40% with respect to Q(r) near the pole. As simulations evolve, the latitudinal differences are smaller. The convective core is prolate, i.e., elongated in the direction of the rotation axis. This feature is consistent with the prediction of the Solberg-Hoiland criterion (6.53). However, this formal core is surrounded by a region of overshooting, which is broader at the equator so that the overall region experiencing convective mixing is about spherical.

The numerical simulations indicate that the analytical approximations used in current stellar models are very rough and that their relative agreement with observational constraints likely results from the freedom offered by their adjustable parameters.

6.5 Convective Envelope in Rotating O-stars

Figure 5.8 indicates that massive O-stars have a small external convective envelope due to their high luminosity. Rotation amplifies these external convective regions. This occurs despite the inhibiting of the Solberg-Hoiland criterion, because another more important effect is present in envelopes: the rotational increase of the radiative gradient Vrad [357]. Let us write the Solberg-Hoiland criterion for stability in the case of a constant p as is usual in envelopes

Vad - Vrad + Vq sin $ > Owith Vq = -H5W ^> (^

ggrav5 W3 dW

where W = r sin $ is the distance to the rotation axis and 5 = - (d ln g/d ln T)P. Here, we specify ggrav and grad. The local flux and the equation of hydrostatic equilibrium are

F = -X^T and VP = qgeff, with x = 4acT3/(3kq). (6.57)

Radiation pressure is included in P, the total pressure. With the von Zeipel theorem (4.21) and (4.22), the local radiative gradient becomes

V = dTdnP = 3 kl(p) p rad dndPT 16 nacGM* (r) T4 , ( )

where the derivatives are taken along a direction n perpendicular to the isobars. Except L(P), the terms are local and thus have to be taken at the given (r, &). With the Eddington factor r (3.117), we get

0 0

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