Dw xH d

2 2 dw1

where f.i = cos d. (The quantities ¡xv and f.iH refer to the spherical model corresponding to e = 0.) The problem of finding the meridional flow (i.e., u1) is thus separated from that of evaluating the reaction of these currents on the overall rotation rate (i.e., w1). In other words, the overall rotation of O(e1/2) forces a small departure from spherical symmetry, which generates large-scale meridional motions of O(e); these, in turn, react back on the driving mechanism, giving rise to differential rotation of O(e3/2).

Correct to O(e), the circulation velocity u can be represented by Eq. (4.28). However, because we have retained turbulent friction in Eqs. (4.50) and (4.51), the functions p12,

Pi2, and Tij2 must be replaced by the following relations:

where G represents the contribution from turbulent friction. Since we must retain the dominant part of the viscous force near the boundaries only, we shall let G = 0 in the bulk of the radiative zone, and we shall write t d ( dv\

near the core-envelope interface and the free surface. Note that this function is nothing but the dominant term //V(due/dr) in Eq. (3.130). By making use of Eq. (4.29), one readily sees that G contains the third-order derivative of the radial function u. Since it is not yet known how to model the variations of //V with any confidence, we shall closely follow the examples set in Eqs. (2.66) and (2.67). Thus, we shall let //V = 10N/rad, where N (> 0) is a constant and /rad is the coefficient of radiative viscosity,

4a T4

15c Kp where k is the coefficient of opacity per unit mass.

Inserting next Eqs. (4.59) and (4.60) into Eq. (4.30), we obtain, after collecting and rearranging terms, vi 4nG3 m3 p3 n - 3/2

where us is defined in Eq. (4.32), and where CVIu is a sixth-order differential operator acting on the function u. Since CVIu = 0 in the bulk of a radiative envelope, we thus recover Sweet's frictionless solution u = us. Near the two boundaries, however, one must explicitly solve Eq. (4.63) together with appropriate boundary conditions (see Section 2.2.2). In particular, we must ensure that matter is flowing along the free surface (see Eq. [4.38]). Moreover, the components of the stress vector acting on the outer boundary, nk (-PSlk + aik), (4.64)

must identically vanish. At the core-envelope boundary, however, the components defined in Eq. (4.64) must be continuous across that boundary. For the sake of simplicity, we shall also prescribe that the core boundary acts as an effective /¿-barrier (although another boundary condition could easily be conceived).* Hence, we shall also apply condition (4.38) at the lower boundary.

* Short of a better theory for the convective core, we have thus assumed that the core is a uniformly rotating, isentropic fluid (i.e., a polytrope of index n = 3/2, which is rotating at the constant angular velocity Q0). Strictly speaking, if convective core overshooting was properly taken into account, one should then solve for both the convective core and the radiative envelope. In practice, however, given some ad hoc description for the overshooting, one could either apply condition (4.38) at a (somewhat larger) effective core radius or prescribe some penetration velocity at the core radius r = Rc.

Near the core boundary, one finds that v 1 d6u

LV u = /xv(n + 1)r2 — + ... = - /xv(n + 1)r2 — + •••. (4.65) dr5 6 dr6

We can also expand the effective polytropic index in the form

where

and i LR

One can easily show that Sc/Rc ^ 1 so that Sc may be taken as a measure of the boundary-layer thickness. Letting next r — Rc Scu x =-c and y = —, (4.70)

Sc v we can rewrite Eq. (4.17) in the form d6 y dx6- xy=-'; (47l)

theorigin x = 0(i.e., r = Rc) becomes, therefore, a simple turning point for the equation. One can also show that our boundary conditions are y = IT = dry = 0 at x = 0. (4.72)

dx dx2

Finally, since the solution ofEq. (4.71) should match the frictionless solution at a distance from the core boundary, we must also have y ^ —, as x (4.73)

Figure 4.1 illustrates the solution ofEq. (4.71) that satisfies conditions (4.72) and (4.73).

In order to discuss the motions in the surface boundary layer, we shall prescribe the usual radiative-zero boundary conditions on the spherical model. Hence, letting z = R — r, we have p = pbzn+1, p = pbzn, T = Tbz, and fxV = 10w ¡xbz. As usual, one has n = 3 for electron-scattering opacity and n = 3.25 for Kramers' opacity law. To exhibit the differences between the core and surface boundary layers, we shall let, without confusion, Fig. 4.1. Function y(x) in the core boundary layer. The frictionless solution, y = 1/x, is indicated by a dashed curve. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49,317, 1982.

where

Again, one has S/R ^ 1 so that S may be regarded as a measure of the boundary layer thickness. It then becomes a simple matter to show that Eq. (4.63) reduces to