The hydrodynamical mechanism

In Section 8.2 we have assumed that an asynchronous binary component goes through a succession of rigidly rotating states, thus overlooking the ability of a gaseous body to develop large-scale currents in meridian planes passing through its rotation axis. The possibility that a secondary circulation controls, in part or in toto, the rotation rate of a tidally distorted star is discussed in this section. Admittedly, this is a comparatively new development that involves specialized concepts in hydrodynamics. In Section 2.5.3 we have already discussed the spin-down of a cyclonic vortex in the Earth's atmosphere, which is the archetype of many situations that are encountered in rotating fluids. For the sake of clarity, in Section 8.4.1 we shall also consider the spin-up and spin-down of an incompressible fluid confined between two parallel infinite plates, when these solid boundaries are subject to an impulsive change in the magnitude of their angular velocity. I shall then explain how to apply these results to the problem of an asynchronously rotating binary component. The transposition is far from being obvious, however, because one has to replace boundary conditions on a solid surface by boundary conditions on a free surface; conditions (2.17) and (2.18) are then replaced by conditions (2.20) and (2.21).

Unless the reader is already familiar with geophysical fluid dynamics, I recommend reading Sections 2.5.3 and 8.4.1, which are essential for the understanding of the doublestar problem treated in Section 8.4.2. Sections 8.4.3 and 8.4.4 present practical applications to detached close binaries; they are almost self-contained and can be read without going through the mathematical derivations made in Sections 8.4.1 and 8.4.2.

8.4.1 The spin-up and spin-down of a rotating fluid

Consider the problem in which two parallel infinite plates and the fluid between them initially rotate with the constant angular velocity Q. The angular velocity of the two plates is then impulsively changed to the new constant value Q0. We wish to describe the manner by which the fluid spins up (when Q < Q0) or spins down (when Q > Q0) to its new angular velocity Q0.

As was shown in Section 2.2.3, the equations describing the motion of a viscous incompressible fluid, in a frame rotating about the z axis with constant angular velocity Q0, are d u 1 2 --+ u ■ grad u + 2Q01z x u = ge--grad p + vV u (8.29)

where u is the velocity in the rotating frame. Remaining symbols have their standard meanings (see Eqs. [2.9] and [2.27]). In cylindrical polar coordinates (m, q>, z), the initial and boundary conditions corresponding to the impulsive change (^ — ^0) in the magnitude of the angular velocity are: u = (^ — ^0)m 19, for t < 0, and u = 0, for t > 0, on the solid plates z = +L and z = —L.

In order to discuss the relative importance of the terms u ■ grad u and 2^01z x u in Eq. (8.29), it is convenient to define the dimensionless ratio e =

which may be described as a Rossby number varying between zero and one (see Eq. [2.30]). For the moment, we shall assume that the initial and final angular velocities differ by a small amount (i.e., e ^ 1), so that the nonlinear terms u ■ grad u can be rightfully neglected in Eq. (8.29). Accordingly, by taking the curl of this equation, one finds that d dt

Equations (8.30) and (8.32) define three scalar equations for the three components of the velocity vector u. This is the so-called linear spin-up (or spin-down) problem. The corresponding nonlinear problem will be discussed in fine.

In the linear approximation, the unsteady solution is of the form

m dm where ^ is the stream function of the large-scale axisymmetric currents and a is a free parameter. As was shown by Greenspan and Howard (1963), boundary-layer theory can be used to solve this problem. Retaining only the highest order derivatives, we obtain d2

d z2

where S = (v/ ^0)1/2 is the boundary-layer thickness. Parenthetically note that one has

where E is the Ekman number of the problem (see Eq. [2.32]).

Here we shall assume that u = (^ — Q0)m for t < 0. The no-slip condition further implies that, at every instant (t > 0), one has dV

Qo L2

at z = +L and z = —L (see Eq. [2.18]). To ensure that the fluid does not penetrate into the solid walls, one must also let, at every instant (t > 0),

Making use of conditions (8.37), one can show that the appropriate boundary-layer solution of Eqs. (8.34) and (8.35) is u = (Q, — Qo)^(1 — e—'% cos %) (8.39)

V = 1 (Qi — Qo) m [az t Se—% (cos % + sin %)]. (8.40)

Here we have defined the stretched variables

near the upper and lower plates, respectively. In Eq. (8.40) the minus sign refers to the boundary-layer solution near the upper plate z = +L and the plus sign to that near the lower plate z = —L.

Conditions (8.38) further imply that we must let

Since Eq. (8.33) has a time dependence of the form exp(—Q0at) - or, equivalently, exp(—t ¡t ) - it follows at once that the e-folding time of the velocity u in the rotating frame is equal to (Q0S¡L)—One thus has f L2 Y¡2

The foregoing linearized problem has been also studied by Greenspan and Howard (1963) using Laplace transforms of Eqs. (8.34) and (8.35). As they showed, the detailed motion consists of three distinct phases: (i) the formation of thin Ekman layers near the two rotating plates, where viscous friction plays a dominant role, (ii) the formation of a large-scale meridional flow that spins up (or spins down) the fluid exponentially, with an e-folding time of the order of td(S/L)—where td is the dynamical time scale and S/L is the relative boundary-layer thickness, and (iii) a much slower decay of the small-amplitude residual motions over the characteristic time of viscous friction, which is of the order of td(S/L)—2.

Figure 8.1 illustrates, at a given instant, the transient meridional flow between two parallel infinite plates that are impulsively spun down. Broadly speaking, the initial impulsive braking of the plates slows down the motion of the fluid, so that a radial inflow of matter takes place within the Ekman layers. By continuity, this radial inflow of matter requires motion along the rotation axis and a slow compensatory outward radial flow in the bulk of the fluid. Since viscous friction is negligible away from the solid walls, this slow outward motion approximately conserves the specific angular momentum Qm2.

Upper Plate

Upper Plate Equatorial Plane

Fig. 8.1. Streamlines of the transient meridional flow in an incompressible fluid between two parallel infinite plates, with the solid walls being spun down impulsively at t = 0 (solid lines). Because the configuration is symmetric with respect to the equatorial plane, the lower half of the fluid is not represented. To illustrate the streamlines near the walls, we have let S/L = 0.05, which corresponds to a rather large viscosity. For comparison, the frictionless solution, which does not satisfy the boundary condition, is also illustrated (dashed lines). Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 395,259, 1992.

Equatorial Plane

Fig. 8.1. Streamlines of the transient meridional flow in an incompressible fluid between two parallel infinite plates, with the solid walls being spun down impulsively at t = 0 (solid lines). Because the configuration is symmetric with respect to the equatorial plane, the lower half of the fluid is not represented. To illustrate the streamlines near the walls, we have let S/L = 0.05, which corresponds to a rather large viscosity. For comparison, the frictionless solution, which does not satisfy the boundary condition, is also illustrated (dashed lines). Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 395,259, 1992.

Accordingly, by replacing high angular velocity fluid by low angular velocity fluid, the large-scale secondary flow serves to spin down the fluid far more rapidly than could mere viscous friction. An entirely analogous, but reverse, phenomenon occurs if the two plates are spun up slightly rather than spun down, but t is then the spin-up time.

It is immediately apparent from these discussions that the spin-up and spin-down times are equal in the linear approximation (i.e., when e ^ 1). A quite different picture emerges when nonlinear effects are taken into account, that is to say, when the restriction to extremely small Rossby number is relaxed. Results for impulsive spin-up and spin-down between parallel infinite plates have been presented by Weidman (1976) for the complete range 0 < e < 1. Of practical interest is the time it takes for the bulk of the fluid to spin up or spin down. In Figure 8.2 we present these two characteristic times - t99, say - in units of the e-folding time t , as functions of the Rossby number. (By definition, they are the elapsed times for which the fluid locally reaches 99% of the change in angular velocity imposed on the solid walls.) Figure 8.2 obviously shows two important features of the problem: (1) The nonlinearity monotonically increases the spin-up and spin-down times and (2) a nonlinear spin-up is achieved somewhat faster than a nonlinear spin-down. Note also that the effects of nonlinearity become of paramount importance as the Rossby number approaches unity, with both characteristic times becoming then much larger than their common value obtained in the linear approximation (see Eq. [8.43]).

8.4.2 Ekman pumping in a tidally distorted star

Following closely the assumptions made in Sections 8.2.1, we have a gaseous star (of mass M, radius R, and luminosity L) acted on by tidal forces originating from a point-mass companion (of mass M ). The primary is assumed to move in a circular orbit Fig. 8.2. Characteristic times for spin-up and spin-down between two parallel infinite plates as functions of the Rossby number (see Eq. [8.31]). Source: Weidman, P. D., J. FluidMech., 77, 685, 1976.

about their common center of mass, with its rotation axis perpendicular to the orbital plane. Let the center of mass of the primary be taken as the origin of our system of spherical polar coordinates (r, d, As usual, the x axis points toward the point-mass companion, and the z axis is parallel to the rotation axis.

If synchronization has not yet been achieved, it is evident that the primary is not at rest with respect to the frame corotating with the orbital angular velocity . To be specific, if is a typical value of the initial rotational angular velocity, then the rotational velocity in the corotating axes is u = (^ — 1V. (In this particular frame, thus, a state of perfect synchronism corresponds to u = 0.) Such a purely azimuthal motion can only be approximate, however, because the primary is always elongated in the direction of the line joining the two centers of mass. This lack of axial symmetry around the rotation axis is illustrated in Figure 8.3, where the four arrows indicate the tidal attraction corrected for the gravitational attraction at the center of mass of the primary. (The small tidal lag is not represented because it plays a negligible role in the Ekman-pumping process. This is an approximation, of course, since it is this tidal lag that will eventually permit a secular exchange of energy and of rotational and orbital angular momenta.) Evidently, if there were no tidal bulges, the motion would remain forever axisymmetric in the corotating frame. Because of the presence of these tidal distortions, however, each fluid parcel in the surface layers is forced to move along an ellipse, in a plane parallel to the equator, with slight accelerations and decelerations along its trajectory. Fig. 8.3. Differential tidal attraction due to the mass point M' (i.e., the tidal attraction corrected for the gravitational attraction at the center of mass of the primary), at four places in the equatorial belt of the primary. The rotation axis is perpendicular to the plane of this schematic drawing. The vertical arrow indicates the sense of the orbital motion. The small tidal lag is not represented.

Fig. 8.3. Differential tidal attraction due to the mass point M' (i.e., the tidal attraction corrected for the gravitational attraction at the center of mass of the primary), at four places in the equatorial belt of the primary. The rotation axis is perpendicular to the plane of this schematic drawing. The vertical arrow indicates the sense of the orbital motion. The small tidal lag is not represented.

It is a simple matter to demonstrate that there exists a class of geostrophic motions that satisfy these requirements. However, one can also demonstrate that these purely azimuthal flows generate a tangential stress vector having a meridional component and a smaller azimuthal component that is proportional to the departure from axial symmetry (see Tassoul and Tassoul 1992, pp. 607-608). Thence, because both components of the stress vector must vanish at a free boundary, one can show that slow but inexorable meridional currents are needed to cancel out its azimuthal component at the surface of an asynchronous rotator. We may anticipate, therefore, that a large-scale meridionalflow will always be required to satisfy all the basic equations and all the boundary conditions, when synchronization has not yet been achieved. As we shall see, these transient currents play a role that is quite similar to that of the secondary flows described in Sections 2.5.3 and 8.4.1.

To formulate the problem in its most general terms, one should solve simultaneously the hydrodynamical equations and the equations that describe the tidal interaction between two deformable bodies. In particular, because the magnitude of the orbital angular velocity is slowly varying in time, the correct form of the equations of motion is

= A - grad f V - - Q20^2 - w) - - gradp + - F(u), (8.44)