Chapter 4 and some parts of Chapter 5 have been devoted to the study of large-scale circulations generated by nonspherical perturbations to the structure of a star. More specifically, Section 4.6 was concerned with the steady, thermally driven meridional flows that result from tidal distortion and mutual heating of the stellar components in a detached close binary. The necessity for mechanically driven currents in an asynchronously rotating binary component was further discussed in Section 8.4.2. As was noted, the importance of these transient motions lies in the fact that they serve to synchronize the axial and orbital motions far more rapidly than could turbulent diffusion of momentum. For completeness, in Section 8.5 we have also presented a qualitative study of the astrostrophic currents that arise from the nonuniform heating at the base of the common envelope in a contact binary. Since each of these four independent flows exhibits quite distinct features, I shall briefly discuss their differences and similarities.

In all studies of meridional circulation in stars, the assumption is made that turbulent friction can be neglected altogether in the bulk of the configuration. It is generally accepted that the flow calculated on the base of a simple frictionless model does provide an adequate representation of the motion at some distance from the boundaries. However, as I have several times mentioned in the book, a frictionless solution does not satisfy the kinematic boundary condition n ■ u = 0 , with |u| finite, (8.89)

neither at the free surface nor at the /¿-barrier defining a core-envelope interface. This is the basic reason why one must retain turbulent friction in a very thin layer of fluid adjacent to each boundary. In this thin boundary layer the normal component of the velocity is diminished continuously from its interior value to a limiting value of zero at the boundary. It is thus the turbulent viscous force, which contains second-order derivatives in the velocity u, that allows both the radial component n ■ u to vanish and the tangential component n x u to remain finite at the boundaries. In other words, the presence of viscous friction increases the order of the equations in the boundary layers so that it is possible to satisfy as many boundary conditions as the basic equations demand (see Section 2.2.2). For some reason, however, this well-known fact has been (and still is) frequently ignored in the astronomical literature.

A good example of the importance of boundary layers in stars is provided by the reflection effect in detached close binaries. Indeed, as was shown in Section 4.6.2, the presence of a "hot spot" on the photosphere of a binary component generates large-scale superficial currents. It is immediately apparent from Figure 4.9 that this axially symmetric circulation remains confined to a thin thermo-viscous boundary layer, with the speed of the flow decreasing exponentially with optical depth. Of course, to this boundary-layer flow we must add the steady circulations generated by rotation and tidal attraction alone, and which are illustrated in Figures 4.3 and 4.8. In each case, the time scale of the thermally driven currents is equal to the Kelvin-Helmholtz time, GM2/RL, divided by a small number that measures the corresponding departure from spherical symmetry (see Eqs. [4.9], [4.127], and [4.136]). As was explained in Section 4.8, these currents are quite different from those encountered in geophysics and laboratory hydrodynamics.

By contrast, there is an evident similarity between the problems treated in Sections 8.4.1 and 8.4.2. (Compare Figures 8.1 and 8.4.) In each case, the motion consists of three distinct phases: the formation of an Ekman-type suction layer near the boundaries, the establishment of a large-scale meridional flow that spins down (or spins up) the frictionless interior, and finally the viscous decay of small residual oscillations.

In spite of obvious differences, these two problems have one essential feature in common, namely, a change in the azimuthal motion near the boundaries. In the laboratory problem, this change is due to the frictional force acting near the solid walls; in the double-star problem, it is caused by the tidal attraction that forces the fluid particles in the surface layers to move nonuniformly along noncircular orbits (see Section 8.4.3). No matter whether the boundaries are solid or free, however, the spin-down results mainly from the conservation of specific angular momentum in the frictionless interior, with the large-scale advection of angular momentum being regulated by the Ekman layer near the boundaries. Actually, the difference between the time scales defined in Eqs. (8.43) and (8.50) can be ascribed to the nature of the pumping mechanism itself: either an impulsive change in the rotation rate of the two parallel infinite plates or the forced lack of axial symmetry in the azimuthal motion of an asynchronously rotating binary component.

To illustrate this point, let us calculate the spin-down times by means of a simple qualitative argument. In each case, the specific angular momentum is essentially preserved in the frictionless interior. Hence, in the linear approximation, a fluid particle with specific angular momentum Q m2 will acquire the lower angular velocity Q0 by moving radially outward the distance

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