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Although stellar rotation has aroused the interest of many distinguished astronomers and mathematicians for almost four hundred years, the theoretical study of the basic physical processes is largely a development of the twentieth century, indeed of the past thirty years or so. In this book I have attempted to present the theory of rotating stars as a branch of classical hydrodynamics, pointing out the differences and similarities between stars and other systems in which rotation is an essential ingredient, such as the Earth's atmosphere and the oceans. Throughout this volume I have thus assumed that the laws governing the internal dynamics of a rotating star are the usual principles of classical mechanics - basically mass conservation, Newton's second law of action, and the laws of thermodynamics. As is well known, one of the reasons why fluid motions in huge natural systems are so complex derives from the fact that the Navier-Stokes equation of motion is inherently nonlinear, so that the superposition of two solutions of a given problem is not necessarily a solution of that problem. In physical terms, this means that it is not possible in general to describe only the largest scale motions in a rotating star, since these flows will almost certainly interact with a whole spectrum of smaller-scale motions. The necessity of incorporating these small-scale, eddylike and/or wavelike motions into the large-scale flows remains as one of the important problems to be solved in astrophysical fluid dynamics.

With very few exceptions, geophysical and astrophysical problems involve motions of such complexity that progress can be made only through a cooperation between formal theory and observation. Since the late 1940s, together with the observations there have been great advances in our theoretical understanding of large-scale phenomena in the Earth's atmosphere and the oceans. By contrast, until the mid-1980s astronomers always had to make use of analytic or numerical models that could not be adequately verified with the available data base. There is little doubt that this lack of direct measurements can explain, at least in part, why the theory of rotating stars is lagging somewhat behind the Earth sciences. It cannot be presented as a complete explanation, however, since prior to the 1970s the oceanographers too had great difficulties in observing the deep interior of the oceans. As a matter of fact, in the first half of the twentieth century, while the geophysicists were assembling the fundamental mechanisms governing the large-scale atmospheric and oceanic flows, the astronomers still had to explain, among other problems, the origin of the energy radiated by the Sun and the stars. Research on stellar interiors thus became primarily a branch of modern physics, with great emphasis being laid on the atomic and nuclear processes. Actually, spherically symmetric stellar models in hydrostatic equilibrium were so successful in accounting for the major observed properties of stars that the most challenging problems of stellar hydrodynamics received comparatively much less attention.

Meridional currents in rotating stars provide a good example of the slow maturation of ideas in stellar hydrodynamics. Indeed, already in 1925, Eddington and Vogt reasoned that the transport of radiation in a rotationally distorted star should cause large-scale motions in meridian planes passing through the rotation axis. Eddington even went a step further, noting that "(these) currents will be deflected east and west by the star's rotation, just as similar currents in our own atmosphere are deflected by the earth's rotation." More importantly, he also wrote: "Presumably when the current has attained a moderate speed a steady state will be reached because the viscosity of the stellar material is considerable and the fundamental equations of equilibrium will be modified by the addition of viscous stresses."* Important advances in our knowledge of the internal dynamics of arotating star were thus made in the 1920s. Yet, despite some far-reaching but incomplete contributions made over the next three decades, the building blocks that explain how the meridional currents and concomitant differential rotation are sustained in an early-type star were not properly assembled until the 1980s - essentially during the period 1982-1995. Chapter 4 presents an overall picture of these techniques, which have been successfully applied to the development of analytic models in which circulation and rotation are represented explicitly but the smaller-scale motions parametrically. Undoubtedly, the most exciting prospects for the future are associated with possibilities of incorporating into numerical models the transfers achieved by these small-scale motions, resolving individual eddy events in sufficient detail to reproduce their transfer properties adequately rather than making use of ad hoc coefficients of eddy viscosity.

Because asteroseismology is still in its infancy, the interior of an upper-main-sequence star has remained so far terra incognita) Hence, there has been as yet little contact between observation and the theoretical studies of large-scale flows in the early-type stars (see, however, Section 6.4). Yet, the accumulation of relatively recent observations has made it clear that, while we understand the fundamentals of stellar evolution, the so-called standard models are in error in a number of details. This is particularly true for early-type stars, and stellar rotation is currently the favorite candidate to explain the discrepancies. Indeed, Herrero et al. (1992) have found that all fast rotators among O-type stars show large surface helium abundances correlated with the rotation rate, which indicates that there is probably a link between rotation and turbulent mixing in these

* Eddington, A. S., The Internal Constitution of the Stars, p. 285, Cambridge: Cambridge University

Press, 1926 (New York: Dover Publications, 1959). t The number of oscillation modes detected in main-sequence stars and white dwarfs is by many orders of magnitude smaller than that in the Sun. This number is much too small to determine the radial structure of a star directly from measured frequencies, as it is done in helioseismology (see Section 1.2.2). Even in the most favorable cases (such as the white dwarf PG 1159-035, in which about one hundred frequencies have been identified with high-order g-modes), the observed splitting of the modes only provides global information about the star's rotation: the value of its rotation period, Prot = 1.35 d, and the evidence that it is rotating nearly uniformly. For a recent survey of asteroseismology, see W. A. Dziembowski, in Sounding Solar and Stellar Interiors (Provost, J., and Schmider, F. X., eds.), I.A.U. Symposium No 181, p. 317, Dordrecht: Kluwer, 1997.

stars.* Unfortunately, as was repeatedly pointed out in Sections 3.6 and 5.4.1, it is not possible at this writing to calculate unequivocally the coefficients of eddy diffusivity in the radiative interior of rotating stars. Accordingly, since the choice of these coefficients is far from being unique, this will necessarily bring about a certain amount of uncertainty in the numerical treatment of stellar evolution. That is to say, despite the fact that rotation has long been known to be capable of inducing turbulent mixing in stellar radiative zones, we are not yet in a position to provide a fully quantitative explanation for the data. Moreover, because the practical evaluation of the eddy diffusivities of matter and momentum in the radiative interior of a rotating star is at least partly an art, not just a science, there is so far no clear expectation for the large-scale flow deep inside an upper-main-sequence star. This could hardly be more different than the situation encountered in late-type star studies, since new observational techniques have recently provided a great deal of information about the internal rotation of the Sun and the rotational evolution of low-mass stars.

Till the late 1980s, theoretical models invariably predicted that the angular velocity in the solar convection zone was constant on cylinders concentric to the rotation axis; moreover, there were then some indications that the Sun's radiative core might be rotating much more rapidly than the surface. According to the most recent helioseismological data, however, it is now generally thought that the rotation rate in the solar convection zone is similar to that at the surface, with the outer parts of the Sun's radiative core rotating uniformly at a rate somewhat lower than the surface equatorial rate. (The rotation rate in the inner core is more uncertain, but recent measurements indicate that these regions might indeed rotate rigidly down to the center.) The 1980s have thus seen our knowledge of the Sun's internal rotation go from the level of mere speculation to that of a field in which the interplay between theory and observation has become indispensable. Yet, it is clear that we are still a long way from an understanding of the interaction between rotation and turbulent convection. Furthermore, because we cannot infer the internal motions of the Sun in a purely deductive manner from the basic equations, our present understanding of its rotational history remains at best phenomenological. As was pointed out in Chapter 5, refined measurements of the Sun's angular velocity in its most central regions will be needed to identify unequivocally the mechanisms that are continuously redistributing the internal angular momentum in response to the rotational deceleration of the solar convection zone.

The 1990s have also witnessed rapid progress in the theoretical study of low-mass stars, both before and during the main-sequence phase. Again, numerical models have provided the opportunity to delve into the component mechanisms responsible for the rotational evolution of these stars, namely, disk-star magnetic coupling during the early phases and internal angular momentum redistribution and saturated magnetized stellar winds during the later phases. It may not be inappropriate to recall, however, that the numerical simulations presented in Chapters 5 and 7 do not "explain" the current observations but rather provide new insights into processes that are not easily explored with the available

* Herrero, A., Kudritzki, R. P., Vilchez, J. M., Kunze, D., Butler, K., and Haser, S., Astron. Astrophys., 261, 209, 1992. For a comprehensive review of these and related matters, see Marc Pinsonneault, "Mixing in Stars," Annu. Rev. Astron. Astrophys., 35, 557, 1997.

instrumentation. This is all the more true in the case of those evolutionary sequences of rotating models that can reproduce the abnormal abundances of the light chemical elements in the Sun and solar-type stars. Indeed, because several adjustable parameters are usually needed to describe the turbulent diffusion processes in the radiative interior of a star, it is quite clear that these models cannot provide the kind of understanding that one would develop from a theory based on first principles alone. Yet, these parameterized models serve a useful purpose because they can be constrained by requiring that the present-day Sun depletes the light elements in the observed proportions, and so they can be used to estimate the gross amount of turbulence present in stellar radiative interiors. Not unexpectedly, the more we progress the more we uncover new, unresolved problems.

In conclusion, it is well to recall that throughout this book I have made use of concepts and methods that were originally introduced in the Earth sciences - barotropy and baroclinicity, geostrophy, eddy-mean flow interaction, boundary-layer theory, etc. In particular, following the example set by the meteorologists and the oceanographers, I have attempted to present consistent solutions that satisfy all the basic equations and all the boundary conditions. This is the reason why we have found that the large-scale motions in the radiative or convective regions of a rotating star always consist of an overall motion around the rotation axis together with much slower but inexorable meridional currents - a situation not unlike those encountered in the Earth's atmosphere and the oceans. In most cases, these secondary flows are dynamically unimportant in the sense that they have little or no effect on the global structure of a rotating star. There is, however, an important exception: the transient meridional currents that advect angular momentum throughout the interior of an asynchronously rotating binary component. As was shown in Chapter 8, this mechanism is closely related to Ekman pumping, and so it is of direct relevance to the study of synchronization and orbital circularization in the close (and not so close) binary stars. This is a fairly new concept in astronomy that was essentially developed between 1987 and 1997; hence, unlike other approaches based on celestial mechanics or resonant interactions with natural modes of oscillation, it has not yet become a part of the astronomical tradition. In this, as in many other debatable issues, it is the accumulation of new observational data that will eventually resolve the controversy. In the present problem, it is essential to improve, observationally, the upper period limits above which detached binaries are asynchronously rotating or have noncircular orbits.

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