It is generally thought that diffusion processes are responsible for most of the peculiar abundances observed in the chemically peculiar stars. As was originally noticed by Michaud (1970), abundance anomalies appear, on the main sequence, in the atmospheres of stars most likely to have stable envelopes and atmospheres. These stars are slow rotators and so have less meridional circulation, they often have magnetic fields, and they have an effective temperature for which stellar envelope models give the weakest convection. In its simplest form, the diffusion model assumes that the region below the superficial convection layer of a chemically peculiar star is stable enough so that microscopic diffusion processes can separate the light elements from the heavy ones, that is to say, those chemical elements absorbing more of the outward going radiative flux per atom move to the surface, while those absorbing less sink into the interior. Because those stars where the thermally driven currents are expected to be the slowest also have the largest abundance anomalies, it is evident that a detailed understanding, from first principles, of the interaction between diffusion and meridional circulation becomes essential if we are to understand stellar abundances.

As was pointed out in Section 4.1, because strict radiative equilibrium is impossible for a uniformly rotating star, a state of thermal equilibrium can only be maintained with the help of energy transport by circulatory currents in meridian planes passing through the rotation axis. In the case of a slowly rotating, early-type star, this large-scale meridional flow is quadrupolar in structure, with rising motions at the poles and sinking motions at the equator (see Figure 4.3). In spherical polar coordinates (r, d, we can thus write u = e dPe(cos e )

u(r ) Pe(cos e )1r — r v(r )sin e —--1e d cos e

where P2 is the Legendre polynomial of degree two and e is the ratio of centrifugal force to gravity at the equator, veq r e = GM - (614)

6 pr2 dr so that the meridional velocity u depends on the radial function u only.

In Table 4.1 we list Sweet's (1950) frictionless solution for a Cowling point-source model. One readily sees that this solution, which becomes infinite at the free surface, does not satisfy the essential boundary conditions (4.38). The situation is even worse when the prescribed rotation law is nonuniform since, for then both components of the meridional velocity become infinite at the free surface (see Eq. [4.42]). In fact, no further progress has been made until it was realized, in 1982, that turbulent friction acting in the outmost surface layers is an essential ingredient of the problem (see Sections 4.3 and 4.4). That is to say, unless one makes allowance for a thermo-viscous boundary layer near the upper boundary of the radiative zone, it is impossible to calculate a meridional flow that satisfies all the boundary conditions and all the basic equations of the problem.+ Table 4.2 lists some of the self-consistent solutions obtained by Tassoul and Tassoul (1982, 1995).

Many characteristics of the chemically peculiar stars can be explained on the basis of microscopic diffusion in the presence of meridional circulation in their outer radiative envelopes. Indeed, when this large-scale flow is rapid enough to obliterate the settling of the diffusion of helium, no underabundance of this element is possible and the superficial He convection zone remains important, making the appearance of some of the abnormal abundances impossible. Comparing the meridional circulation velocities for solid-body rotation (i.e., a = 0 in Table 4.2) to diffusion velocities of helium below the He convection zone, Michaud (1982) has shown that this zone disappears only in stars with equatorial velocities smaller than about 90 km s-1. This is in agreement with the

* There has been much confusion in the literature about the existence of thermo-viscous boundary layers in rotating stars. This is discussed in the Bibliographical notes for Section 4.3.

cutoff velocity observed for the HgMn stars. Given this encouraging result, detailed two-dimensional diffusion calculations have been carried out by Charbonneau and Michaud (1988) to determine with greater accuracy the maximum rotational velocity allowing the gravitational settling of helium.

In order to couple microscopic diffusion and meridional circulation, one writes the continuity equation in the form d c p--+ div [p(u + U)c] = div(pD12 grad c), (6.16)

d t where c(r, d, t ) is the concentration of the contaminant, measured with respect to hydrogen, and D12(r ) is the coefficient of diffusion of helium in hydrogen. The velocity field u(r, d) corresponds to the meridional circulation, while U(r, d, t) describes the advec-tive part of the diffusion velocity (e.g., Charbonneau and Michaud 1988, pp. 810-811). Equation (6.16) is a parabolic equation that must be solved with appropriate initial and boundary conditions. Calculations have been performed in both 3M0 and 1.8M0 stellar models appropriate, respectively, for HgMn and FmAm stars. The upper limits to the equatorial velocities allowing the chemical separation of helium are found to be 75 and 100 km s-1, respectively, for these stars. Given the various approximations that had to be made in averaging over convection zones and the uncertainties in the meridional circulation velocities near the surface, the agreement with observations is quite satisfactory. This parameter-free model is not so successful, however, in reproducing quantitatively the anomalies of a given star in detail. Mass loss has been suggested as an important ingredient in the FmAm phenomenon.

Now, because turbulent particle transport can also have drastic effects on chemical separation, Charbonneau and Michaud (1991) have performed additional calculations that retain both meridional circulation and anisotropic turbulence. Equation (6.16) was thus replaced by d c p--+ div[p (u + U)c] = div(p D grad c), (6.17)

d t where the total diffusivity tensor can be written in the form

(Compare with Eq. [3.134].) The functions DV and DH are the vertical and horizontal coefficients of eddy diffusivity due to the rotationally induced instabilities. Unfortunately, as was explained in Section 3.6, there is as yet no reliable theory that could provide firm analytical expressions or numerical values for these two coefficients. They are essentially free quantities that must be chosen, by trial and error, using the observed abundance anomalies to determine their values. Thus, given some parametric expressions for the eddy diffusivities, the problem can be treated as a two-dimensional initial-boundary value problem. Numerical calculations show that the diffusion model for FmAm stars is particularly constraining regarding the introduction of anisotropic turbulence. In the presence of meridional circulation, it is found that the maximum DV/D\2 ratio tolerable with the diffusion model is of the order of 10; otherwise, helium settling is overly impeded in stars rotating below the observed equatorial velocity cutoff. This sets extremely tight constraints on turbulence in early-type stars having equatorial velocities of 100 km s-1

or less. Similar calculations show that the maximum DH/D12 ratio tolerable with the diffusion model for FmAm stars is of the order of 106; otherwise, helium settling remains possible in stars rotating above the observed equatorial velocity cutoff. As was pointed by Charbonneau and Michaud (1991), however, this seems to be a prohibitively large value of the ratio DH/D12.

In summary, the above calculations show that microscopic diffusion in the presence of large-scale meridional currents does explain in a natural way the appearance of the HgMn and FmAm phenomenon in slowly rotating, nonmagnetic stars, without introducing any strong dependence on arbitrary parameters. These calculations also demonstrate that the smaller-scale, eddylike motions cannot be ignored altogether because they, too, can impede the gravitational settling of helium. In principle, given some solution for the meridional circulation, one can integrate Eq. (6.17) to derive upper limits on the coefficients DV and DH. As was pointed out in Section 4.4, however, the topology of the meridional flow in the surface layers of an early-type star is quite dependent on the gradient of angular velocity in these regions. Since this uncertainty on the circulation pattern should somewhat reflect on the determination of upper limits on DV and DH, it follows that the relative importance of meridional circulation and anisotropic turbulence in reducing chemical separation remains uncertain.

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