Equatorial Plane

Fig. 5.6. Same as Figure 5.5, but at t & 5 Gyr (curve 6 of Figure 5.3). Note the virtual disappearance of meridional currents in the domain r¡ R& < 0.40. Source (revised): Tassoul, M., and Tassoul, J. L., Astrophys. J.., 279, 384, 1984.

of an evolved solar model. In the chemically homogeneous part of the radiative core, however, the typical time scale of the rotationally driven currents remains of the order of the Eddington-Sweet time, tES = R/e\uQ\^(GM2/RL)/e (see Eq. [4.37]). Typically, because e ^ 10-4 and \ua \ ^ 10-5 cm s-1 in the outer part of the Sun's radiative core, one finds that tES > 1012 yr, which is much larger than the Sun's age.

Now, as was pointed out at the end of Section 4.3.1, ew1 describes the back reaction of the meridional flow on the basic rotation rate w0 (see Eq. [5.22]). It is a simple matter to show that one has e \w1/w0\ ^ e\puar /iV \ ^ tV / tES,where iV is the vertical coefficient of eddy viscosity and tV is the viscous time scale. If we let iV = 10Nim, where im is the microscopic viscosity and N is a positive number, detailed numerical calculations indicate that one has e\w1/w0\ ^e 104-N. Since e ^ 10-4inthe Sun, one readily sees that a moderate amount of turbulence (N ^ 2-3, say) is amply sufficient to ensure that, to a first approximation, the viscous friction acting on the mean azimuthal flow dominates over the advection of specific angular momentum by the rotationally driven currents.

5.4 Spin-down of the solar interior

As we shall see more closely in Section 7.2, the Sun and solar-type stars sustain a continuous loss of mass as a result of magnetized stellar winds and/or episodic mass ejections emanating from their outer convection zones. The rotational deceleration of the convective envelope that results from the application of this torque leads to the creation of internal stresses that act to redistribute angular momentum within the radiative core. Yet, as was pointed out in Section 1.2.2, analyses of helioseismological data strongly suggest that from the base of the convection zone down to r ^ 0.1-0.2R0 the Sun's interior is rotating at a rate close to that of the surface equatorial belt, while the inner core is perhaps rotating more rapidly than the chemically homogeneous parts of the radiative interior. Because angular momentum is continuously transferred away from the surface convection zone to outer space, it follows at once that there must exist a very effective mechanism of angular momentum transport inside the Sun, thus keeping the bulk of the radiative interior rotating approximately uniformly in spite of the inexorable solar-wind torque.

Broadly speaking, if we describe the mean velocity field as the sum of an overall rotation and a large-scale meridional flow, three mechanisms can redistribute angular momentum within the Sun's radiative interior: (i) the advection of specific angular momentum by the meridional currents, (ii) the diffusion of momentum arising from turbulent friction acting on the differential rotation, and (iii) the interaction with a large-scale magnetic field. Specifically, making use of Eq. (5.1), one can write the y component of the momentum equation in the form d 2 2 p — (Qm ) + pu ■ grad(Qm ) d t

where m = r sin d and v is the kinematic viscosity (see Eq. [4.146]). The vectors Hp and Hy are, respectively, the poloidal and toroidal parts of the magnetic field.

In Section 5.3 we have shown that the typical speed \u\ of the thermally driven meridional currents is so slow that, to a first approximation, the advection of angular momentum by these currents can be neglected in Eq. (5.40). In fact, two categories of models have been proposed. In one of them, angular momentum redistribution is treated as a turbulent diffusion process, with advection by the meridional flow and magnetic fields being neglected altogether. The other group of models is based on the idea that this redistribution is dominated by magnetic stresses arising from the shearing of a preexisting poloidal magnetic field. It is to these two distinct approaches that we now turn.

5.4.1 Rotation and turbulent diffusion

It has long been recognized that standard evolution theory is quite successful in explaining the main properties of stars. Yet, as more data become available, the limits of the standard spherical models have become more apparent. For example, the observed solar lithium abundance is afactorof200 smaller than that found in meteorites, indicating that some downward particle transport has occurred in the outer parts of the Sun's radiative core. As was noted by Endal and Sofia (1981), rotation might be the ultimate cause of this slow mixing process, since rotationally induced instabilities will generate a wide spectrum of small-scale motions that produce internal mixing of certain chemical species.

Following these authors, the mechanisms that redistribute angular momentum can be divided into two categories, dynamical and thermal, according to the time scales associated with the triggering mechanisms. Hence, whenever the shear-flow and symmetric instabilities arise in their models, the angular velocity gradient is instantaneously readjusted to a state of marginal stability by radial exchange of angular momentum (see Eqs. [3.93] and [3.101]).* However, because of the longer time scales for the thermal instabilities, the overall redistribution of angular momentum and chemical composition are computed using the coupled diffusion equations:

for the mass fraction Xi (r, t) of chemical species i. The function D, which is sensitive to both angular velocity and chemical composition gradients, is the coefficient of eddy viscosity due to the rotationally induced thermal instabilities. (It was denoted by v in Eq. [5.40].) Note that these equations may be derived at once from Eqs. (3.133) and (3.134), assuming that the ratio of eddy diffusivity to eddy viscosity is equal to the constant f. As usual, the eddy coefficient D is taken as the product of some typical length Lc and some typical speed Vc, which is assumed to be the sum of velocities generated by the Eddington-Sweet currents and some thermal instabilities. As was noted in Section 3.6, however, such a formulation is at best phenomenological because it is not yet known how to model the variations of the function D with any confidence. In fact, because the eddy coefficients cannot be calculated from first principles alone, their

for the angular velocity Q(r, t), and

* Parenthetically note that the ever-present barotropic and baroclinic instabilities discussed in Section 3.4.3 are not taken into account in these models.

overall magnitude can be determined only by adjusting the constant f and the empirical formula for the function D to the observational constraints.

Several evolutionary models that include the combined effects of rotationally induced mixing and angular momentum redistribution in the Sun's radiative core have been calculated by Pinsonneault, Kawaler, Sofia, and Demarque (1989). Following current practice, the effects of rotation were treated as small distortions superimposed on spherically symmetric models (see Section 6.2). For some reason, however, Eq. (5.41) was replaced by

where I is the moment of inertia and M is the mass of the Sun. In some calculations, Eq. (5.42) was also modified to include the combined effects of rotationally induced mixing and microscopic diffusion. Following Chaboyer, Demarque, and Pinsonneault (1995), we thus have

where Dm1 and Dm 2 are derived from the microscopic diffusion coefficients and multiplied by the adjustable parameter fm. As usual, these equations must be supplemented by appropriate initial and boundary conditions. In particular, one must prescribe some general expression for the continuous loss of angular momentum due to the magnetically coupled solar wind.

The evolutionary models have been calibrated to match the usual global properties of the present-day Sun, as well as its observed rotation rate. Numerical calculations indicate that the value of f is approximately 0.033. This result is in perfect agreement with the fact that turbulent diffusion of matter is a much less effective process than turbulent diffusion of momentum in a stably stratified system (see Section 3.6). Note also that those models that include rotation and microscopic diffusion have convection zone depths of 0.710R0, providing a good match to the observed depth.

As far as rotation is concerned, the models have an oblateness in agreement with the observed upper limit. This is a consequence of a general feature of these models, namely, that they all rotate slowly in the outer layers where the contribution to oblateness is greatest. Angular momentum transport in the models is also remarkably efficient in smoothing out differential rotation in the radiative core. The possible range of rotation profiles for models with angular momentum transport is compared to a model with the same surface rotation velocity but without transport in Figure 5.7. Note that the rotation curve for r > 0.6R0 is almost flat in the models. Inside the radius r = 0.6R0, however, the degree of differential rotation depends on the choice of parameters. Now, as was noted in Section 1.2.2, inversion of the available ^-mode oscillation data suggests a nearly flat rotation curve down to r ^ 0.1-0.2R0. Accordingly, it appears most likely that a more efficient angular momentum transport mechanism is present in the Sun - one that is not present in the models developed by the Yale group.

At this juncture it is appropriate to mention the work of Schatzman (1996), who pointed out that gravity waves generated by turbulent stresses in the solar convection zone might also contribute to the almost uniform rotation of the Sun's radiative interior. Original calculations by Kumar and Quataert (1997) and others show that there is enough angular

Fig. 5.7. Angular velocity as a function of radius in the present-day Sun for three distinct models of angular momentum transport. (w is the angular velocity.) The solid line is the rotation curve the present-day Sun would have if it started with an average initial angular momentum and evolved to the age of the Sun without any transport of angular momentum from the radiative interior to the surface convection zone. The long-dashed line is a model with very inefficient angular momentum transport. The short-dashed line is a model with very efficient angular momentum transport. Source: Pinsonneault, M. H., Kawaler, S. D., Sofia, S., and Demarque, P., Astrophys. J., 338,424, 1989.

Fig. 5.7. Angular velocity as a function of radius in the present-day Sun for three distinct models of angular momentum transport. (w is the angular velocity.) The solid line is the rotation curve the present-day Sun would have if it started with an average initial angular momentum and evolved to the age of the Sun without any transport of angular momentum from the radiative interior to the surface convection zone. The long-dashed line is a model with very inefficient angular momentum transport. The short-dashed line is a model with very efficient angular momentum transport. Source: Pinsonneault, M. H., Kawaler, S. D., Sofia, S., and Demarque, P., Astrophys. J., 338,424, 1989.

momentum in gravity waves generated by convection that they can force the outer parts of the radiative interior into corotation with the base of the convection zone in about 107 yr. Even though these results are dependent on the description chosen for the turbulent motions in the solar convection zone, they clearly show that turbulent diffusion due to random gravity waves is a physical process that cannot be ignored.

5.4.2 Rotation and magnetic fields

In Section 5.4.1, the models for the evolution of the internal solar rotation have been computed assuming angular momentum transport solely by hydrodynamical means. In this section we shall investigate the rotational deceleration of a solar model containing a large-scale poloidal magnetic field in its radiative core, in response to the torque applied to it by a magnetically coupled wind. The first quantitative study of this problem was made by Charbonneau and MacGregor (1992). Their investigation was conducted using a numerical model that includes treatment of both convection zone braking by the magnetized solar wind and internal angular momentum redistribution by magnetic and viscous stresses.

For the sake of simplicity, we shall neglect the meridional velocity u in Eq. (5.40), and we shall assume strict axial symmetry for the large-scale magnetic field. Equation (5.40) then becomes dr. - 1

p — (Om2) = div(pvm2 grad O) + —- Hp ■ grad(mHy), (5.45)

91 4n where the poloidal magnetic field Hp (r, d) is assumed to be time independent and known a priori. With these simplifications, the spin-down problem reduces to solving Eq. (5.45) and the y component of the induction equation,

where p is the magnetic diffusivity (see Eq. [4.148]). Both v and p are assumed to be constant throughout the radiative core, with the adopted value for v being small enough that viscous transport of angular momentum is negligible compared to magnetic transport. Such a formulation is self-consistent because it takes into account (i) the generation of the toroidal component Hy (r, d, t) by shearing of the poloidal field and (ii) the back reaction on the angular velocity O(r, d, t) due to the nonvanishing Lorentz force associated with the time-varying toroidal component of the magnetic field. When supplemented by some initial and boundary conditions, Eqs. (5.45) and (5.46) describe a two-dimensional problem for the two unknown functions, O(r, d, t) and Hy(r, d, t), governed by two coupled, linear, quasi-hyperbolic equations.

A large set of calculations have been performed by Charbonneau and MacGregor (1993), starting on the zero-age main sequence from a state of solid-body rotation at 50 times the present solar rate and zero toroidal field. They identify two distinct regions in the interior: a convective envelope, which they assume to rotate as a solid body at all times at the rate OCE(t), and an underlying radiative core. The solutions were computed for four distinct poloidal field configurations, as shown in Figure 5.8, and for poloidal field strengths B0 of 0.01, 0.1, 1, and 10 G. Note that the fields D1 and D2 are such that direct magnetic coupling exists between the convective envelope and the radiative core, while for the fields D3 and D4 the envelope is magnetically decoupled from the underlying core.

These spin-down calculations enable us to draw a detailed picture of the magnetic and rotational evolution of an internally magnetized solar-type model, which is acted upon by the torque associated with a magnetically coupled wind. The evolution can be divided into three more or less distinct phases: an initial phase of toroidal field buildup, lasting between a few thousand to a few million years, depending on the topology and strength of the internal poloidal field; a second period in which large-scale toroidal oscillations set up in the radiative core during the first phase are damped; and a third period, lasting from age of about 107 yr onward, characterized by a state of dynamical balance between the total stresses (magnetic plus viscous) at the base of the convective envelope and the wind-induced surface torque, leading to a quasi-static internal magnetic and rotational evolution.

The time evolution of internal differential rotation is shown in Figure 5.9. The dimen-sionless quantity AO is constructed by integrating the difference O(r, d, t) — aCE(t) over the magnetized part ofthe radiative interior, thus providing a global measure ofthe difference in angular velocity between the convective envelope and the magnetized part of the

Fig. 5.8. Poloidal magnetic field configurations. The bottom of the convective envelope is located at RCE = 0.74302R&. Source: Charbonneau, P., and MacGregor, K. B.,Astrophys. J., 417, 762, 1993.

radiative core. Note that in all cases AQ initially increases very rapidly before reaching a maximum at about t ^ 108 yr. At later times, AQ declines at a rate nearly independent of poloidal field strength and configuration. An important common property of these solutions is the weak differential rotation that most of them exhibit by the time they have attained the solar age. Except for the D4 configurations, all solutions have AQ < 0.02

Fig. 5.9. Time evolution of the internal differential rotation, as defined in the text by the quantity AQ, for various poloidal field configurations and strengths. In (A) are shown solutions for the four poloidal configurations of Figure 5.8, all at a strength of 1 G. In (B), (C), and (D) are shown the effects of varying the poloidal field strength for a given poloidal configuration. Source: Charbonneau, P., and MacGregor, K. B.,Astrophys. J., 417, 762, 1993.

Fig. 5.9. Time evolution of the internal differential rotation, as defined in the text by the quantity AQ, for various poloidal field configurations and strengths. In (A) are shown solutions for the four poloidal configurations of Figure 5.8, all at a strength of 1 G. In (B), (C), and (D) are shown the effects of varying the poloidal field strength for a given poloidal configuration. Source: Charbonneau, P., and MacGregor, K. B.,Astrophys. J., 417, 762, 1993.

at t = 4.5 x 109 yr. This is in contrast to the unmagnetized models, which often have significant angular velocity gradients in their radiative cores even at the solar age.

These quantitative studies are important because, for the very first time, they demonstrate the existence of classes of large-scale internal magnetic fields that can accommodate rapid spin-down of the surface layers near the zero-age main sequence and yield a weak internal differential rotation in the radiative core by the solar age. The lack of significant differential rotation from the base of the solar convection zone down to r ^ 0.1-0.2R0 would then exclude from further consideration poloidal magnetic configurations of the D4 type. Within the current observational uncertainties, all D1, D2, and D3 solutions are compatible with the results reported in Section 1.2.2. However, none of these solutions exhibit enhanced angular velocity inside the radius r = 0.2R0, as some helioseismolog-ical observations have suggested. Following Charbonneau and MacGregor (1992), this can be achieved by choosing poloidal magnetic fields such that the inner core remains magnetically decoupled from the surrounding regions. Admittedly, there is no firm justification for such a choice, but it seems to be the only way to have a rapidly rotating inner core (if any) in the present-day Sun.

More recently, Rudiger and Kitchatinov (1996) have performed a large set of spin-down calculations, making allowance for the differential rotation in the convective envelope (see Section 5.2.1). Their work thus combines differential rotation at the base of the solar convection zone, rotational braking due to a magnetically coupled solar wind, and an axially symmetric magnetic field in the Sun's radiative interior. A reasonable picture emerges only if the following two conditions are met: (1) viscosity is strongly enhanced compared to its microscopic value, and (2) the internal magnetic field does not penetrate into the outer convection zone. As was shown in Section 4.7.2, an axially symmetric poloidal magnetic field makes the rotation uniform along each field line, although the constant angular velocity is in general different for each field line. If the internal poloidal field was anchored into the differentially rotating convective envelope, the latitudinal rotation inhomogeneity would thus penetrate deep into the radiative interior, which is not observed. With the magnetic field fully embedded into the core, however, their models do reproduce the thin layer where a transition from differential to rigid-body rotation occurs at the bottom of the solar convection zone. (This transition layer is known as the solar tachocline.) The problem is then presented by the "dead zone" permeated by the field lines that never come close to the base of the convective envelope (see the D3 and D4 configurations in Figure 5.8). This is the reason why a sizable amount of eddy viscosity is needed to link this region to the base of the solar convection zone across the magnetic field. To be specific, the models of Rudiger and Kitchatinov (1996) require an eddy viscosity of the order of 104 ¡m, where ¡¡m is the microscopic viscosity (see also the end of Section 5.3). As they noted, however, there is no contradiction at this point with the models of Charbonneau and MacGregor (1993), since these solutions also require an amplification factor of the order of 104 in the coefficient of viscosity.

In summary, two independent sets of spin-down calculations have been made. They differ in one important respect, however. In the Charbonneau-MacGregor models the convective envelope is assumed to rotate uniformly at all times, whereas the latitudinal differential rotation of that zone is properly retained in the Rudiger-Kitchatinov models. In both sets of models, it is found that there exist large-scale magnetic fields that yield a weak internal differential rotation by the solar age. In the former case, however, the poloidal field lines may or may not penetrate into the convective envelope (see Figure 5.8). By contrast, in the latter case, the helioseismological observations are reproduced only with the poloidal magnetic field fully contained within the radiative core. In both cases, the models are quite insensitive to the magnitude of the internal magnetic field, provided the poloidal field strength B0 is larger than 10-3 G. However, despite the high efficiency of these magnetic fields in transporting angular momentum, turbulent friction is always needed to enforce almost uniform rotation in the radiative interior by the solar age.

5.5 Discussion

In Sections 4.3 and 4.4 we have presented a simple but adequate description of the mean state of motion in a nonmagnetic early-type star that consists of a uniformly rotating convective core and a surrounding radiative envelope. Assuming no mass loss from the star's surface, we have shown that there exists a mean steady solution for the large-scale motion in the radiative envelope, which is the combination of an overall differential rotation and slow circulatory currents in planes passing through the rotation axis. As was pointed out, however, the major impediment to the complete resolution of this problem is the lack of quantitative observational data about the velocity field in the surface layers of an early-type star. This is in contrast to the late-type stars, as a variety of recent observational results have shed important new light on both the internal rotation of the Sun and the rotational evolution of solar-type stars. It is therefore appropriate at this juncture to critically review the degree of development of the main theories of solar rotation.

It is generally believed that the interaction of rotation with convection plays an essential role in the generation and maintenance of differential rotation and concomitant meridional circulation in the solar convection zone. Unfortunately, although it has been suggested that rotation may be interacting with either local turbulent convection or global turbulent convection, no scheme has yet been generally accepted as being basically correct. In fact, because a general theory of turbulent convection still lies in the distant future, in all likelihood further progress will result from a balanced approach that involves increasingly reliable helioseismological observations combined with more and more sophisticated numerical simulations.

Considerable progress has been made in determining the processes that affect the internal rotation of the Sun. In Section 5.3 we have shown that thermally driven meridional currents inexorably advect angular momentum in the chemically homogeneous parts of the Sun's radiative core, thus tending to induce small departures from solid-body rotation in these regions. For the rotation rate of the present-day Sun, this large-scale advection of angular momentum is probably negligible, although in a more detailed study it might effectively contribute to the angular momentum redistribution within the outer parts of the Sun's radiative core.

Two very efficient mechanisms for angular momentum redistribution in the solar interior have been thoroughly investigated: turbulent friction acting on the differential rotation and large-scale magnetic fields. As was shown in Section 5.4, both of them provide the means by which the solar-wind torque is communicated to the interior, while enforcing almost uniform rotation in the radiative core by the solar age.

In the turbulent models illustrated in Figure 5.7 the angular momentum redistribution within the Sun's radiative core is treated diffusively. As was repeatedly pointed out, this approach is, at best, a semiquantitative one (see, e.g., Section 3.6). Indeed, by making use of the crude concept of eddy viscosity, one necessarily relegates all eddy and/or wave events to a passive means of dissipating the large-scale flow, thus implying an ill-defined energy cascade from the largest to the smallest scales of motion. And because one cannot calculate the eddy coefficients from first principles alone, it follows that one must integrate Eqs. (5.41) and (5.42) under widely different conditions, thence guessing the form and values of the empirical formula for the function D that best fit the global properties of the present-day Sun. Note also that these turbulent models, which often have significant angular velocity gradients in their radiative cores even at the solar age, are generally characterized by the presence of a small, rapidly rotating central core. Such a behavior is attributable to the fact that the development with age of a gradient of mean molecular weight in the hydrogen-burning core leads to a much reduced eddy viscosity in these parts of the solar interior, thus preventing them from participating to the overall redistribution of angular momentum.

In Section 5.4.2 we have shown that a more efficient means for transporting angular momentum in the Sun's radiative core is through the intermediary of a large-scale internal magnetic field. Detailed numerical simulations demonstrate the existence of classes of poloidal fields allowing rapid surface spin-down at early epochs, while producing almost uniform rotation throughout the Sun's radiative core by the solar age. However, these calculations show that a certain amount of a turbulent friction is always required to couple the field lines. They also indicate that the observed surface rotation rate is a rather poor indicator of the strength and geometry of hypothetical large-scale magnetic fields pervading the solar radiative regions. As far as the internal rotation is concerned, the most important property of these models is the weak overall differential rotation that most of them exhibit by the time they have attained the solar age. This is in contrast to the diffusive models presented in Section 5.4.1, which exhibit enhanced angular velocity in their central regions r < 0.2R0. Since the actual rotation rate inside this radius is still very uncertain, we are therefore led to the conclusion that the relative importance of the two basic mechanisms for angular momentum redistribution deep inside the Sun is also an open question.

5.6 Bibliographical notes

Comprehensive introductions to the Sun are:

1. Stix, M., The Sun, Berlin: Springer-Verlag, 1989.

2. Foukal, P., Solar Astrophysics, New York: John Wiley and Sons, 1990.

Sections 5.1 and 5.2. The concept of anisotropic eddy viscosity was originally applied to the solar rotation problem by Lebedinski (Reference 31 of Chapter 3). The reference to Weiss is to his paper:

3. Weiss, N. O., The Observatory, 85, 37, 1965. The first detailed mean-field models are due to:

5. Durney, B. R., and Roxburgh, I. W., Solar Physics, 16, 3, 1971.

Subsequent mean-field models are reviewed in:

6. Stix, M., in The Internal Solar Angular Velocity (Durney, B. R., and Sofia, S., eds.), p. 392, Dordrecht: Reidel, 1987.

7. Rüdiger, G., Differential Rotation and Stellar Convection, New York: Gordon and Breach, 1989.

More details about the Kitchatinov-Rüdiger models will be found in:

8. Kitchatinov, L. L., and Rüdiger, G., Astron. Astrophys., 299, 446, 1995. Observational constraints on the solar rotation theories have been discussed at length in:

9. Dürney, B. R., Astrophys. J., 378, 378, 1991; ibid, 407, 367, 1993.

10. Chiü, H. Y., and Paterno, L., Astron. Astrophys., 260, 441, 1992.

The main properties of the global-convection models devised by Gilman and Glatzmaier are sümmarized in:

12. Gilman, P. A., and Miller, J., Astrophys. J. Suppl., 61, 585, 1986.

13. Glatzmaier, G. A., in The Internal Solar Angular Velocity (Dürney, B. R., and Sofia, S., eds.), p. 263, Dordrecht: Reidel, 1987.

Fülly türbülent regimes are considered in:

14. Glatzmaier, G. A., and Toomre, J., in Gong 94: Helio- and Astero-Seismology (Ulrich, R. K., Rhodes, E. J., Jr., and Dappen, W., eds.), A.S.P. Conference Series, 76, 200, 1995.

Papers of related interest are:

15. Pülkkinen, P., Tüominen, I., Brandenbürg, A., Nordlünd, A., and Stein, R. F., Astron. Astrophys., 267, 265, 1993.

16. Canüto, V. M., Minotti, F. O., Schilling, O., Astrophys. J., 425, 303, 1994.

17. Brummell, N. H., Xie, X., and Toomre, J., in Gong 94: Helio- and Astero-Seismology (Ulrich, R. K., Rhodes, E. J., Jr., and Dappen, W., eds.), A.S.P. Conference Series, 76, 192, 1995.

18. Brümmell, N. H., Hürlbürt,N. E., and Toomre, J., Astrophys. J., 473,494,1996; ibid., 493, 955, 1998.

19. Vandakürov, Yü. V., Astronomy Letters, 23, 55, 1997.

Very little is known aboüt the solar tachocline, that is, the thin velocity boündary layer below the convection zone, where there exists an ünresolved transition to almost üniform rotation. This and related matters are treated in:

20. Spiegel, E. A., and Zahn, J. P., Astron. Astrophys., 265, 106, 1992.

21. Gilman, P. A., and Fox, P. A., Astrophys. J., 484, 439, 1997.

22. Rüdiger, G., and Kitchatinov, L. L., Astron. Nachr., 318, 273, 1997.

Other papers may be traced to Reference 23 (p. 216) of Chapter 1; see also:

23. Basü, S.,Mon. Not. R. Astron. Soc., 288, 572, 1997.

Section 5.3. The inhibiting role of a /¿-gradient was first pointed oüt by Mestel (Reference 9 of Chapter 4). Qüantitative stüdies will be foünd in:

24. McDonald, B. E., Astrophys. Space Sci., 19, 309, 1972.

25. Huppert, H. E., and Spiegel, E. A., Astrophys. J., 213, 157, 1977.

The time-dependent models reported in this section were originally obtained by:

26. Tassoul, M., and Tassoul, J. L., Astrophys. J., 279, 384, 1984.

The effect of a stellar-wind torque on the meridional flow was further discussed in:

27. Tassoul, M., and Tassoul, J. L., Astrophys. J., 286, 350, 1984.

Section 5.4.1. These diffusive models have their roots in the work of Endal and

Sofia:

28. Endal, A. S., and Sofia, S., Astrophys. J., 243, 625, 1981. Detailed evolutionary models have been reported in:

29. Pinsonneault, M. H., Kawaler, S. D., Sofia, S., and Demarque, P., Astrophys. J., 338, 424, 1989.

30. Chaboyer, B., Demarque, P., and Pinsonneault, M. H., Astrophys. J., 441, 865,

1995.

An illustration of their empirical coefficient D will be found in Reference 40 (Figure 16, p. 548) of Chapter 3. Compare with the results obtained by:

31. Tassoul, J. L., and Tassoul, M., Astron. Astrophys., 213, 397, 1989. Angular momentum transport by gravity waves has been discussed in:

33. Kumar, P., and Quataert, E. J., Astrophys. J. Letters, 475, L143, 1997.

The efficiency of this transport mechanism has been confirmed independently by:

34. Zahn, J. P., Talon, S., and Matias, J., Astron. Astrophys., 322, 320, 1997.

35. Charbonneau, P., and MacGregor, K. B., Astrophys. J. Letters, 397, L63, 1992.

36. Charbonneau, P., and MacGregor, K. B., Astrophys. J., 417, 762, 1993.

Their original results have received confirmation in the following work:

37. Kitamaya, O., Sakurai, T., and Ma, J., Geophys. Astrophys. FluidDyn., 83,307,

1996.

Solar spin-down models that include differential rotation in the convective envelope are due to:

38. Rudiger, G., and Kitchatinov, L. L., Astrophys. J., 466, 1078, 1996.

Section 5.5. A detailed comparison between theory and helioseismological observations of the Sun's internal angular velocity profile will be found in:

39. Charbonneau, P., Tomczyk, S., Schou, J., and Thompson, M. J., Astrophys. J., 496,1015, 1998.

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