where f is to be interpreted as the coefficient of magnetic eddy diffusivity in the turbulent radiative envelope. For the sake of simplicity, we shall assume that f = ff T-v, where f3 and v (>0) are two constants. As usual, these equations must be combined with Eqs. (4.3)-(4.6).

Because we are considering a rotating magnetic star that does not greatly depart from spherical symmetry, the large-scale meridional flow is the linear superposition of rotationally driven currents and magnetically driven currents. To calculate these currents, we shall prescribe an axially symmetric dipolar field. Neglecting the circulation velocity and letting H = P in Eq. (4.148), we thus have, in spherical polar coordinates (r, L = cos d, y ),

2 dPiL)

with pm(R) = 1 so that H is the initial polar field strength. The constant ct is the lower eigenvalue of

4 ct

P'm + ~Pm + J TVPm = 0, (4.152) r p with RPm+ 3Pm = 0 at r = R, and Pm finite at r = 0. Given this large-scale magnetic field, we shall now expand about hydrostatic equilibrium in powers of the nondimensional parameter

H R4

To the decaying dipolar field P corresponds the following meridional velocity:

u = Xu(r, t)P2(p)lr + Xv(r, t)(1 - p2)—^ 1„, (4.154)

d p where u = um (r )exp(-2ct t) and v = vm (r )exp(-2ct t). (4.155) As usual, we also have

6 pr2 dr and 2P2(p) = 3p? — 1. (Recall that ue = — ru^/sin d.)

Following Section 4.3.1, boundary-layer theory was used to calculate the functions um and vm. Extensive numerical results have been obtained for a Cowling point-source model embedded into a vacuum (M = 3M0, R = 1.75Re, L = 93L0), with electron-scattering opacity and with fzV = 105^rad in the boundary layers. Equations (4.9) and (4.153) become, therefore, e = 3 x 10-6 ue2q and X = 9 x 10-17 H2, (4.157)

where the equatorial velocity veq (= R) and the constant H are measured in km s-1 and gauss, respectively. (Letting veq = 60 km s-1 and H = 103 G, one has e ^ 10-2 and X ^ 10-10.) Now, solving Eq. (4.152) for the decay time tp (= a-1), one finds that 3tp = 4.2 x 1023 (when v = 1.5) and ¡tp = 5.4 x 1037 (when v = 3.5). For example, if we neglect turbulence altogether, 3 becomes equal to its ideal value vm = 1013 T-3/2 cm2 s-1 so that one has tp = 4 x 1010 yr. Obviously, shorter decay times can be obtained by choosing other values for the free parameters 3 and v; these times must be compared with the main-sequence lifetime tms of a 3M0 star, which is of the order of 2 x 108 yr.

As was already noted, correct to the orders e and X, the large-scale meridional flow is the vectorial sum of rotationally driven currents and magnetically driven currents. Henceforth we shall call them the "^-currents" and the "H-currents," respectively. Table 4.5 lists the functions u, rv, um, and r vm, in cgs units, when v = 1.5 and v = 3.5. All entries in the last four columns must be multiplied by the exponential factor exp(-2t¡xp); they do not depend on ¡. Even when H is as large as 103-104 G, one readily sees that the steady ^-currents are much faster than the slowly decaying H-currents in the bulk of the radiative envelope. (At r = 0.6R, one has |ur | ^ 10-6 cm s-1 for the currents, whereas |ur | ^ 4 x 10-10 cm s-1 at t = 0 for the H-currents). Just below the surface, however, Xum may become larger than e u, in spite of the fact that both um and u vanish at the top of the boundary layer (i.e., at r = R). The presence of sizable H-currents in the outermost surface layers of a magnetic star is not at all unexpected, since it is only in the low-density surface regions that the Lorentz force can generate sizable departures from spherical symmetry. Figure 4.10 illustrates the complex circulation pattern of the H-currents; correct to O(X), it does not depend on the polar field strength. Figure 4.10 must be compared with Figure 4.3, which depicts the corresponding ^-currents.

From Table 4.5, one readily sees that the values of um and vm are quite sensitive to the exponent v, that is to say, to the magnitude of the coefficient of magnetic eddy diffusivity. Thus, even though it is the eddy viscosity that ultimately prevents unwanted singularities in the circulation velocities at the surface, the role of the magnetic eddy diffusivity is nevertheless an essential one in the sense that it considerably reduces the magnitude of these velocities near the surface. Unless one makes the unrealistic demand that the motions be strictly laminar in a chemically homogeneous, fully ionized radiative envelope, there is no reason to select the value v = 1.5, however. This should be especially true because hydrogen is only partially ionized at the surface of many magnetic stars, thus increasing the diffusion coefficient 3.

Now, because the ^-currents and the H-currents are neatly separated to the orders e and X, these solutions can be used also to obtain the circulation pattern when the axis of the basic dipolar field is inclined at an angle x to the rotational axis. Because we already know that the H-currents play a negligible role in the bulk of a radiative envelope, we shall

Fig. 4.10. Lines of force of the dipolar magnetic field (left) and streamlines of the corresponding quadrupolar circulation (right), when v = 3.5. The shape of these curves does not depend on the polar field strength. Recall that the dipolar field decreases as exp(-1¡xp) whereas the meridional currents decrease as exp(-2t ¡xp). Note also the accumulation of streamlines near the core boundary and near the free surface. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 310, 786, 1986.

Fig. 4.10. Lines of force of the dipolar magnetic field (left) and streamlines of the corresponding quadrupolar circulation (right), when v = 3.5. The shape of these curves does not depend on the polar field strength. Recall that the dipolar field decreases as exp(-1¡xp) whereas the meridional currents decrease as exp(-2t ¡xp). Note also the accumulation of streamlines near the core boundary and near the free surface. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 310, 786, 1986.

merely follow in time the distortions of an initially dipolar field that may be caused by the Q-currents alone. Three-dimensional calculations show beyond any doubt that the slow but inexorable Q--currents will indeed convert an initially inclined dipolar field into a more complex field that has a larger inclination over the rotation axis. Figure 4.11 illustrates the evolution of an initially dipolar field, with initial x = 45°, e = 10-3,and Tp = 2 x 108yr. (The rotation axis is set to be vertical.) Accordingly, assuming a modest increase of the coefficient ft over its ideal value vm (fi/vm ^ 102) and choosing a rotation that is typical for a magnetic star (e ^ 10-3), we have shown that the Q-currents are by far too inefficient to produce a perpendicular rotator over the main-sequence lifetime of a 3M0 star. In other words, because the field lines can more easily diffuse through a less-than-ideal body, one has random orientation of the axes, whereas in an idealized stellar model (with ft = vm) one has an excess of perpendicular rotators. Since the observed distribution of the obliquities seems to be at most a marginal nonrandom one, these calculations corroborate the idea that small-scale, eddylike motions comprise an essential ingredient of the many theoretical problems that are raised by rotation and magnetism in a stellar radiative envelope.

4.7.2 Circulation, rotation, and magnetic fields

In Section 4.7.1 we have calculated the meridional flow in the radiative envelope of a rotating magnetic star, assuming that departures from spherical symmetry are not too large. As we know, these rotationally and magnetically driven currents advect angular momentum and, hence, interact with the azimuthal motion. In the absence of a magnetic field, the transport of specific angular momentum can be made to balance the viscous force arising from differential rotation so that the mean state of motion is a steady one (see Eqs. [4.55]-[4.57]). When a large-scale magnetic field pervades the system, however, this balance is modified by the presence of the toroidal component of the Lorentz force. This is the reason why the claim has often been made that there exists a weak, axially

v = |
1.5 |
v= |
3.5 | ||||

r/ R |
u |
r v |
um |
r Vm |
um |
r Vm | |

0.283182 |
0. |
0. |
0. |
0. |
0. |
0. | |

0.283400 |
1.340E- |
1 |
4.716E+1 |
3.384E+01 |
1.193E+04 |
1.637E+3 |
5.754E+5 |

0.283600 |
2.347E- |
1 |
-3.983E+0 |
5.940E+01 |
-9.815E+02 |
2.862E+3 |
-4.984E+4 |

0.283800 |
1.640E- |
1 |
-1.935E+1 |
4.160E+01 |
-4.888E+03 |
1.997E+3 |
-2.364E+5 |

0.290000 |
1.355E- |
2 |
-9.628E-2 |
3.633E+00 |
-2.427E+01 |
1.554E+2 |
-1.178E+3 |

0.300000 |
5.414E- |
3 |
- 1.594E-2 |
1.579E+00 |
-4.031E+00 |
5.620E+1 |
-1.949E+2 |

0.350000 |
1.414E- |
3 |
-1.125E-3 |
5.852E-01 |
-2.463E-01 |
8.697E+0 |
-1.218E+1 |

0.400000 |
9.577E- |
4 |
-4.370E-4 |
5.428E-01 |
-2.027E-02 |
3.601E+0 |
-3.838E+0 |

0.450000 |
8.798E- |
4 |
-2.739E-4 |
6.869E-01 |
1.244E-01 |
2.185E+0 |
-1.893E+0 |

0.500000 |
9.566E- |
4 |
-2.347E-4 |
1.050E+00 |
3.644E-01 |
1.720E+0 |
-1.213E+0 |

0.550000 |
1.154E- |
3 |
-2.717E-4 |
1.843E+00 |
9.096E-01 |
1.652E+0 |
-9.324E-1 |

0.600000 |
1.484E- |
3 |
-4.149E-4 |
3.633E+00 |
2.328E+00 |
1.867E+0 |
-8.139E-1 |

0.650000 |
1.983E- |
3 |
-7.627E-4 |
8.022E+00 |
6.485E+00 |
2.435E+0 |
-7.586E-1 |

0.700000 |
2.699E- |
3 |
-1.533E-3 |
2.014E+01 |
2.060E+01 |
3.656E+0 |
-6.545E-1 |

0.750000 |
3.695E- |
3 |
-3.187E-3 |
5.970E+01 |
7.924E+01 |
6.414E+0 |
- 1.207E-1 |

0.800000 |
5.044E- |
3 |
-6.745E-3 |
2.249E+02 |
4.060E+02 |
1.373E+1 |
2.833E+0 |

0.850000 |
6.824E- |
3 |
- 1.472E-2 |
1.249E+03 |
3.306E+03 |
3.953E+1 |
2.293E+1 |

0.900000 |
9.116E- |
3 |
-3.480E-2 |
1.435E+04 |
6.315E+04 |
1.928E+2 |
2.604E+2 |

0.950000 |
1.200E- |
2 |
- 1.062E-1 |
1.030E+06 |
1.014E+07 |
3.419E+3 |
1.288E+4 |

0.986000 |
1.461E- |
2 |
-5.019E-1 |
4.058E+09 |
1.052E+12 |
1.097E+6 |
4.772E+7 |

0.990000 |
1.418E- |
2 |
-7.772E-1 |
8.503E+10 |
1.040E+12 |
4.637E+6 |
-5.647E+7 |

0.995000 |
8.944E- |
3 |
-1.154E+0 |
1.937E+11 |
-2.036E+13 |
6.421E+6 |
-7.520E+8 |

0.999000 |
1.872E- |
3 |
-1.246E+0 |
5.452E+10 |
-3.622E+13 |
1.482E+6 |
-9.882E+8 |

1.000000 |
0. |
-1.255E+0 |
0. |
-3.646E+13 |
0. |
-9.838E+8 |

Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 310, 786, 1986.

symmetric magnetic field that can offset the advection of specific angular momentum and so keeps the rotation effectively uniform in space, with little or no turbulent motions in the radiative envelope. It is the purpose of this section to conduct an examination of the ways a large-scale magnetic field can indeed maintain almost uniform rotation in the radiative envelope of an early-type star (see also Section 5.4.2).

Let us first assume that the magnetic field is symmetric about the rotation axis. Expressing the mean velocity v and the mean magnetic field H as the sum of poloidal and toroidal parts, v = u + Qm 1y and H = P + T m 1y, (4.158)

we can thus write the y components of Eqs. (4.146) and (4.148) in the forms

Fig.4.11. Evolution of an initially dipolar magnetic field, with x = 45°, e = 10-3,and Tp = 2 x 108 yr. The rotation axis is vertical. The lines of force, which do penetrate the convective core, are depicted in the radiative envelope only: at t = 5 x 104 yr, at t = 4.2 x 107 yr, at t = Tp, and in the asymptotic limit t ^ <x. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J.., 310, 805, 1986.

Fig.4.11. Evolution of an initially dipolar magnetic field, with x = 45°, e = 10-3,and Tp = 2 x 108 yr. The rotation axis is vertical. The lines of force, which do penetrate the convective core, are depicted in the radiative envelope only: at t = 5 x 104 yr, at t = 4.2 x 107 yr, at t = Tp, and in the asymptotic limit t ^ <x. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J.., 310, 805, 1986.

-+ div(Tu) = P ■ grad ^--|curl[^ curl(Tm 19)]V (4.160)

In principle, we must calculate the functions u, P, and T from Eqs. (4.146) and (4.148) and the auxiliary equations (4.3)-(4.6). Following standard practice, however, we shall assume some plausible forms for the circulation velocity u and the poloidal magnetic field P. Accordingly, it is possible to calculate the functions ^ and T from Eqs. (4.159) and (4.160), with u and P being two prescribed vectors. As usual, these coupled parabolic equations must be solved with some initial conditions and a prescribed set of boundary conditions at the core boundary r = Rc and at the free surface r = R. For the sake of simplicity, all numerical calculations reported below were made for a spherical fluid shell with constant density p, constant kinematic viscosity v (= Pv/p), and constant magnetic diffusivity f3. The convective core is assumed to be maintained in strict uniform rotation. Two extensive sets of numerical calculations have been made.

Tassoul and Tassoul (1989) have considered a quadrupolar magnetic field, vanishing at the core-envelope interface and at the free surface, and a prescribed quadrupolar meridional circulation. Specifying initial states of uniform or almost uniform rotation, they obtained solutions that are characterized by an inexorable approach to a state of isorotation (i.e., rotation with constant angular velocity along each field line) with large differential rotation between field lines after about ten Alfven times, with no apparent trend toward solid-body rotation.*

Moss, Mestel, and Tayler (1990) have considered a dipolar magnetic field, fully anchored into the convective core and threading the free surface, and a quadrupolar meridional circulation. They also introduced a high-viscosity buffer zone above the core-envelope interface, in which, however, they retained low values for the magnetic diffu-sivity. Starting from a state of almost uniform rotation, they obtained solutions in which there is a periodic, low-amplitude shear reversal about a state of uniform rotation, along with spatially extended latitudinal oscillations in the toroidal magnetic field.

How can one explain the differences between these two independent sets of calculations? Recall first that, in both works, the magnetic field is symmetric about the rotation axis so that the magnetic transport of angular momentum in the radiative envelope takes place along but not across the poloidal field lines. Accordingly, if these lines thread neither the free surface nor the core-envelope interface, viscous friction is the only mechanism that can potentially couple different field lines. Hence, the redistribution of angular momentum takes place along the field lines through the propagation of Alfven waves. Since ohmic dissipation acts to damp out these waves, it will thus enforce a constant angular velocity along each poloidal field line, although this constant is in general different for each field line. This solution corresponds to the state of isorotation that was obtained by Tassoul and Tassoul (1989). Obviously, this is quite different from the situation in which the poloidal field lines are anchored into the rigidly rotating core. If so, then, there is a coupling between the convective core and the radiative envelope, so that significant mutual coupling of different poloidal field lines will occur. As was noted by Moss, Mestel, and Tayler (1990), it is the anchoring of all poloidal field lines in a rigidly rotating, strongly viscous convective core that is ultimately responsible for the establishment of a state of almost uniform rotation, on a time shorter than the main-sequence lifetime of an early-type star. Of course, if the core is not rotating as a solid body or if some poloidal field lines do not penetrate into the core, the large-scale poloidal magnetic field will not necessarily enforce almost uniform rotation throughout the radiative envelope.

It is appropriate at this juncture to briefly discuss the work of Charbonneau and MacGregor (1992), who have studied the rotational evolution of an initially non-rotating radiative envelope, following an impulsive spin-up of the core to a constant angular velocity. This was accomplished by solving Eqs. (4.159) and (4.160) for a given axially symmetric vector P, neglecting all fluid motions other than the azimuthal flow

* The concept of isorotation - as opposed to solid-body rotation - has its roots in the work of Ferraro (1937) and Mestel (1961).

associated with the evolving differential rotation (i.e., letting u = 0 in these equations). For fully core-anchored poloidal field configurations, they found that uniform rotation is always enforced on a time very much shorter than the main-sequence lifetime, yet generally much larger than the core-to-surface Alfven transit time. However, they also found that the relatively rapid transition toward uniform rotation depends critically on all poloidal field lines having at least one footpoint anchored on the rigidly rotating core. This is well illustrated by their unanchored and partially anchored solutions, which in many cases either do not attain solid-body rotation or do so on a purely viscous time scale.

We can only conclude from these three sets of calculations that the extent to which a weak poloidal magnetic field can produce a state of almost uniform rotation in a stellar radiative envelope depends critically on assumptions regarding the behavior of the field lines at the core-envelope interface. To be specific, if the convective core is not maintained in strict uniform rotation, or if all field lines are not fully anchored into the core, the configuration does not converge toward a state of almost uniform rotation in the radiative envelope. Given our almost complete ignorance of the state of motion in a convective core and of whether all poloidal field lines do penetrate into the core of an early-type star, there is thus no reason to claim that there always exists an axially symmetric magnetic field that can enforce almost uniform rotation despite the inexorable advection of angular momentum by the meridional currents. Such a magnetic field may or may not exist, depending on the field-line topology and rotation in the convective core.

Of course, as was noted by Moss (1992) and others, if the poloidal magnetic field is not symmetric about the rotation axis, nonuniform rotation will generate magnetic torques that can interchange angular momentum between poloidal field lines. Preliminary calculations have been made when the magnetic axis is perpendicular to the rotation axis, suggesting that the azimuthal magnetic forces do indeed establish almost uniform rotation in the radiative envelope of a perpendicular rotator. In my opinion, no firm conclusion can be made until independent studies present reliable calculations of a large number of fully three-dimensional models having arbitrary inclinations. This is another way of saying that, contrary to an often held belief, the presence of a large-scale magnetic field does not make the problem of stellar rotation any simpler.

Self-gravitation and self-generated radiation are the two main factors that make most problems of stellar hydrodynamics quite different from those encountered in the geophysical sciences and in laboratory hydrodynamics. Self-gravitation acts as the "container" of a star, making its outer surface free rather than solid. Hence, it is self-gravitation that allows for the small departures from spherical symmetry - regardless of whether their ultimate cause is the centrifugal force, the Lorentz force, or the tidal interaction with a companion. Moreover, as explained in this chapter, it is the transport of self-generated radiation in anonspherical configuration that causes the slow but inexorable currents and concomitant differential rotation in a stellar radiative zone.

In the case of a slowly rotating, early-type star, the large-scale meridional flow is quadrupolar in structure, with rising motions at the poles and sinking motions at the equator. Typically, the time scale of these thermally driven currents in the bulk of a radiative envelope is equal to the Kelvin-Helmholtz time divided by the ratio of centrifugal force to gravity at the equator (see Eq. [4.37]), which is known as the Eddington-Sweet time.

The complexity of the problem derives from the fact that turbulent friction becomes of paramount importance near the core-envelope interface and the free surface. To be specific, near each of these boundaries, a thin layer exists in which turbulent processes allow the velocity to make the transition from the value required by the nature of the boundary to the value that is appropriate to the interior, frictionless solution. Simultaneously, the frictional force acting on the mean azimuthal flow can be made to balance the transport of angular momentum by the large-scale meridional flow, therefore ensuring that all three components of the momentum equation are properly satisfied.

As far as hydrodynamics is concerned, perhaps the most challenging feature of these motions is that they bear no relation whatsoever to the large-scale circulation encountered in geophysics.

For example, as was seen in Section 2.5.1, for large-scale atmospheric motions away from the Earth's surface the balance is essentially geostrophic (see Eq. [2.79]). On the contrary, one readily sees that Eq. (4.57) defines the balance between the turbulent viscous force acting on the mean azimuthal motion and the inexorable transport of angular momentum by the thermally driven currents. When written in a rotating frame of reference, this equation merely states that the Coriolis force acting on the meridional circulation balances the azimuthal viscous force. In other words, the concept of geostrophy does not apply to the thermally driven currents in a nonspherical stellar radiative envelope.

A comparison between the results obtained in Sections 2.5.2 and 4.3.1 also shows that the boundary layers in a stellar radiative envelope are definitely not of the Ekman type. To be specific, because the meridional flow is essentially caused by the nonspherical part of the temperature field, these boundary layers are of the mixed thermo-viscous type (see Eq. [4.60]). That is to say, whereas turbulent friction plays a dominant role in the energy equation, the mechanical balance is mainly between the pressure-gradient force and the effective gravity; the viscous force is very small in the equations of motion themselves. These boundary layers are also of a singular nature because it is not possible to obtain the boundary-layer solutions by merely adding thermo-viscous corrections to the interior, frictionless solution (see Eqs. [4.71], [4.76], and [4.100]). To the best of my knowledge, there is no equivalent in any other field.

Admittedly, in Section 4.3 we have made use of steady state models to represent the largest scale of motion in a stellar radiative envelope, while applying parametric expressions to describe the effects of all smaller scales. These solutions are basically very similar to the linear and nonlinear solutions that were obtained for oceanic boundary currents: Bryan's nonlinear solution smoothly reduces to Munk's linear solution as the eddy viscosity is gradually increased (see Section 2.6.3). More recently, because it has been established that mid-ocean eddies (~ 50 km) are prevalent, their models have been superseded by high-resolution models that include this eddy field within the large-scale oceanic circulation. In principle, a similar improvement could be made in the case of a stellar radiative zone, taking into account the smaller scales of motion. Unfortunately, very little is known about the intensity, length and time scales, and the spatial distribution of these transient motions. At this writing, it is therefore quite difficult to resolve the eddy field and, at the same time, the large-scale flow in the radiative envelope of a rotating star.

4.9 Bibliographical notes

Sections 4.1 and 4.2. The existence of meridional currents was originally suggested by:

1. Vogt, H., Astron. Nachr., 223, 229, January 1925.

2. Eddington, A. S., The Observatory, 48, 73, March 1925.

Their time scale was discussed in:

3. Eddington, A. S.,Mon. Not. R. Astron. Soc., 90, 54, 1929.

4. Sweet, P. A.,Mon. Not. R. Astron. Soc., 110, 548, 1950.

Equation (4.37) was properly derived in Reference 4. The expansion method is due to:

5. Milne, E. A.,Mon. Not. R. Astron. Soc., 83, 118, 1923. A systematic study was made by:

6. Chandrasekhar, S., Mon. Not. R. Astron. Soc., 93, 390, 1933 (reprinted in Selected Papers, 1, p. 183, Chicago: The University of Chicago Press, 1989).

Other pioneering contributions to the problem of meridional circulation were made by:

7. Gratton, L., Mem. Soc. Astron. Italiana, 17, 5, 1945.

8. Opik, E. J.,Mon. Not. R. Astron. Soc., 111, 278, 1951.

9. Mestel, L.,Mon. Not. R. Astron. Soc., 113, 716, 1953.

10. Baker, N., and Kippenhahn, R., Zeit. Astrophys., 48, 140, 1959.

11. Mestel, L., in Stellar Structure (Aller, L. H., and McLaughlin, D. B., eds.), p. 465, Chicago: The University of Chicago Press, 1965.

13. Smith, R. C.,Mon. Not. R. Astron. Soc., 148, 275, 1970.

Section 4.3. The importance of viscous friction was already noted in Reference 2. This idea was further studied in:

The analysis in this section is taken from:

16. Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49, 317, 1982.

17. Tassoul, J. L., and Tassoul, M., Astrophys. J., 264, 298, 1983.

18. Tassoul, M., and Tassoul, J. L., Astrophys. J., 271, 315, 1983.

19. Tassoul, J. L., and Tassoul, M., Geophys. Astrophys. FluidDyn., 36, 303,1986.

Adequate boundary-layer analyses, at the core and at the surface, were originally made in Reference 16.

As far as the surface boundary layer is concerned, the claim has been made that one can construct a nonsingular surface solution that satisfies all the boundary conditions, without requiring any boundary layer; see:

21. Sakurai, T., Geophys. Astrophys. FluidDyn., 36, 257, 1986.

22. Zahn, J. P., Astron. Astrophys., 265, 115, 1992.

As was shown in Reference 20, neglecting the viscous force acting on the meridional flow, one can obtain a solution that satisfies all the boundary conditions at the free surface. Unfortunately, this nonsingular solution does not satisfy all the basic equations - one component of the momentum equation remains necessarily unfulfilled. As a matter of fact, the Sakurai-Zahn approach is inadequate because it cannot take into account the mathematical singularity of the equations at the free surface (see Eq. [4.100], which has a pole at x = 0). It is also shown in Reference 20 that the dynamics in a slowly rotating, early-type star demands some frictional forces acting on the meridional flow to be present. This requires the consideration of singular, thermo-viscous boundary layers, both at the core and at the free surface.

Section 4.4. Our most detailed study of meridional circulation will be found in:

23. Tassoul, M., and Tassoul, J. L., Astrophys. J., 440, 789, 1995.

This paper contains many numerical illustrations, as well as several comments on the current literature.

Section 4.5. See:

24. Tassoul, M., and Tassoul, J. L., Astrophys. J., 267, 334, 1983.

Section 4.6. See:

25. Tassoul, J. L., and Tassoul, M., Astrophys. J., 261, 265, 1982; ibid, p. 273. The reflection effect in a nonsynchronous binary component was considered by:

26. Tassoul, M., and Tassoul, J. L., Mon. Not. R. Astron. Soc., 232, 481, 1988. The reference to Hosokawa is to his paper:

27. Hosokawa, Y., Sci. Rep. Tohoku Univ., Ser. I, 43, 207, 1959.

Section 4.7.1. Magnetically driven currents are discussed in:

28. Tassoul, J. L., and Tassoul, M., Astrophys. J., 310, 786, 1986.

The effects of meridional streaming on an oblique rotator have been considered by:

29. Tassoul, J. L., and Tassoul, M., Astrophys. J., 310, 805, 1986.

30. Moss, D.,Mon. Not. R. Astron. Soc., 244, 272, 1990.

Section 4.7.2. Since the early 1950s, rumor had it that there exists a weak poloidal magnetic field that is symmetric about the rotation axis and which can quickly enforce almost uniform rotation in a stellar radiative envelope, with little or no turbulent friction, despite the inexorable advection of angular momentum by the meridional currents. In Reference 28, as a sideline to the actual calculation of these currents, we expressed serious doubts about the existence in general of such a field. Not unexpectedly, we had opened Pandora's box, letting out the following sequel of papers:

31. Mestel, L., Moss, D. L., and Tayler, R. J., Mon. Not. R. Astron. Soc., 231, 873, 1988.

32. Tassoul, M., and Tassoul, J. L., Astrophys. J., 345, 472, 1989.

33. Moss, D. L., Mestel, L., and Tayler, R. J.,Mon. Not. R. Astron. Soc., 245, 559, 1990.

Reference 31 presents a set of numerical calculations based on Eqs. (4.159) and (4.160), the results of which are summarized by the following claim: "in both the analytical and numerical treatments above, we have always calculated the ^-field, showing the departures from uniformity to be small" (p. 883). In order to ascertain the universal validity of that statement, similar calculations were made in Reference 32, prescribing in Eqs. (4.159) and (4.160) a poloidal field P that is not anchored in the convective core. It was found that the configuration tends toward a state of isorotation that has a large gradient in the angular velocity near the rotation axis. Reference 33 presents additional calculations, recognizing the difference that exists between anchored and unanchored poloidal field lines. Since unanchored field lines do not necessarily enforce almost uniform rotation, it is now conjectured that a magnetic field that is not symmetric about the rotation axis is most likely to quickly establish almost uniform rotation in a stellar radiative envelope.

Other contributions are due to:

34. Charbonneau, P., and MacGregor, K. B., Astrophys. J., 387, 639, 1992.

35. Moss, D.,Mon. Not. R. Astron. Soc., 257, 593, 1992.

See especially Reference 34, which also contains a useful comparison between their own results and those presented in References 31-33.

The references to Ferraro and Mestel are to their papers:

36. Ferraro, V. C. A.,Mon. Not. R. Astron. Soc., 97, 458, 1937.

37. Mestel, L.,Mon. Not. R. Astron. Soc., 122, 473, 1961.

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