Parenthetically note that condition (4.88) is not automatically satisfied if iV vanishes at the surface, since this also implies that Eq. (4.87) has a singular point at r = R. In fact, Eq. (4.87) has a first integral that is quite convenient for our purposes. Setting the constant of integration equal to zero, one obtains the nonlinear equation dw

therefore ensuring that boundary condition (4.88) is satisfied provided the product pu vanishes at r = R.

A second relation between u and w can be obtained from Eqs. (4.3)-(4.6) and (4.50)-(4.51). In this case, Eq. (4.58) remains valid but Eqs. (4.59) and (4.60) must be replaced by

and where

1 GM p dw2

Again inserting these relations into Eq. (4.30), we obtain

Assuming electron-scattering opacity, one has

2Lr4

The function h is governed by the following inhomogeneous equation:

(Note that f = 0 in the convective core, where one assumes that Î2 = Œ0.) As explained in Section 4.2.1, when solving this equation one must always ensure the continuity of gravity across the core-envelope interface and across the free surface. One also has d2 f

df dr

Equations (4.89), (4.93), and (4.96) form a coupled system for the functions u, w, and h. Away from the boundaries, turbulent friction acting on the meridional flow is negligible so that one can replace Eq. (4.93) by u = uS + uf (4.98)

in the bulk of the radiative envelope. Near the core boundary, one can solve Eq. (4.93) along the lines presented in Section 4.3.1 (see Figure 4.1). Near the free surface, however, one readily sees from Eqs. (4.94)-(4.97) that the frictionless solution uS + uf behaves as 1/p. Following closely Eqs. (4.74)-(4.80), we shall thus let

S (pu f )r in the surface boundary layer. With this new definition for y, Eq. (4.93) becomes y d~y - x2"-2y = -x"-2. (4.100)

Note that Eq. (4.100) is very similar in structure to Eq. (4.76), with x"-2 merely replacing x2"-2 on the right-hand side. Conditions (4.77)-(4.79) remain unchanged but Eq. (4.80) must be replaced by y ^ -1, as x (4.101)

since the solution of Eq. (4.100) should match the frictionless solution at the bottom of the surface boundary layer. Figure 4.4 illustrates the solution of Eq. (4.100) that satisfies conditions (4.77)-(4.79) and (4.101). A uniformly valid solution of Eqs. (4.89), (4.93), and (4.96) can thus be obtained, all the way from the outer boundary to the core-envelope interface.

The above formulation corresponds to the case for which one has tH ^ tES so that we can let O. = Í2(r). As explained in Section 4.3.1, if one also assumes that tV ^ tES, the function O remains nearly equal to a constant. In that case, correct to order e, Eq. (4.86) can be rewritten in the form

After linearizing Eq. (4.89), we obtain

0.08

0.06

0.04

0.02

Fig. 4.4. Function y(x) in the surface boundary layer. The frictionless solutions, y = 1 /xn, are indicated by dashed curves. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49,317, 1982.

0.06

0.04

0.02

Fig. 4.4. Function y(x) in the surface boundary layer. The frictionless solutions, y = 1 /xn, are indicated by dashed curves. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49,317, 1982.

This equation must be solved with the condition ¡31 (Rc) = 0 so that £ = £0 at the core boundary r = Rc. (Condition [4.88] is automatically satisfied since one has pu = 0 at r = R.) This is a major simplification because it implies that f = uf = 0; the right-hand side of Eq. (4.96) is thus identically equal to zero. This, in turn, implies that Eq. (4.90) no longer depends on rotation. Hence, the function u can be calculated along the lines presented in Section 4.3.1. Thence, one can solve Eq. (4.103) to obtain the function jii. This is exactly the problem presented in Eq. (4.81), neglecting of course the d dependence of the function w1. Indeed, by making use of Eq. (4.29), one can easily show that the derivative of Eq. (4.103) is strictly equivalent to Eq. (4.82).

In Section 4.3.1 we have calculated the meridional velocity u and the angular velocity £ in a slowly rotating star, when the departures from solid-body rotation are uniformly small throughout the whole radiative zone. The circulation pattern consists of a single cell (or gyre) extending from the convective core boundary to the free surface, with interior upwelling at the poles that is compensated by interior downwelling at the equator (see Figure 4.3). Although turbulent friction acting on the circulation is negligible in the bulk of the radiative envelope, there exist thin layers in which turbulent friction prevents the formation of unwanted singularities near the inner and outer boundaries. Such boundary-layer solutions satisfy all the basic equations and all the boundary conditions, with the circulation velocities remaining uniformly small throughout the radiative envelope. Of course, in the boundary layers these velocities depend on the coefficient ¡¡V. Fortunately, because they depend, respectively, on (¡¡V)1/7 and (¡V)1/10 in the core and surface boundary layers, their dependence on this poorly known parameter is considerably reduced (see Eqs. [4.68] and [4.75]).

Now, as was noted in Section 4.2.2, the claim has been made that there always exist two distinct cells separated by the level surface with density p = p* (say) given by

22 = 2n Gp *. Moreover, it has been shown that in a frictionless, differentially rotating star one always has ur a 1/p in the surface layers, thus leading to much larger surface velocities than Sweet's (1950). Both objections require that we retain the second-order terms in Eqs. (4.55) and (4.56). However, because differential rotation plays an essential role in the discussion, we shall also replace Eq. (4.55) by

where w0 is a function of the coordinates and time. Following Section 4.3.1, we shall consider the case for which one has tH ^ tV ^ tES, thus ensuring the convergence of expansion (4.104).

The general strategy is as follows. First, one solves to O(e1/2) the y component of the momentum equation for the large-scale motion. Neglecting the d dependence, we obtain p— = —— [iiyr4 — + kyr 3w0 , (4.105)

dt r4 dr V dr where we have retained the XV effect (see Eq. [3.133]). Thus, unless the parameter XV identically vanishes, the solution of Eq. (4.105) does not correspond to a solid-body rotation. For steady motions, we have

Since pV and XV are poorly known quantities, we shall merely prescribe that

where a is a constant. Second, one calculates the first-order velocity ui, which can be obtained from Eqs. (4.28) and (4.93), replacing w by w0 in definition (4.92). Third, once the problem has been solved to that order, one calculates the back reaction w1 (see Eq. [4.81]). Finally, collecting all the pieces together, one calculates the second-order velocity u2. To this order of approximation, however, one must retain the inertial terms u1 ■ grad u1 in the poloidal part of the momentum equation. Correct to O(e2), one has p = p0(r) + e [p1,0(r) + pu(r) Pz(p)]

+ e 2[p2,0(r) + P2,2(r) P2(p) + P2,4(r) P4P)] (4.108)

and similar expressions for p, T, and V. (Henceforth we shall omit the subscripts "0" from the function p0.) With the help of Eq. (4.4), we can also describe the meridional flow by means of a stream function. Thus, we let

pr2 dp pr2 dr (One also has u6 = — rup/sin d.) To the same order of approximation, one finds that dP2(p)

where we have defined the following functions:

It is to be noted that, to O(e2), the streamlines ^ = constant do depend on e, whereas they are independent of this small parameter in the first-order approximation. Correct to O(e2), the angular velocity can be brought to the form

where the y s are governed by a set of inhomogeneous equations.

In Table 4.2 we list the first-order functions u and r v (in cm s-1) for three values of a, in a Cowling point-source model with electron-scattering opacity (M = 3M0, R = 1.75R0, L = 93L0, and N = 6 in the boundary layers). Evidently, the case a = 0 corresponds to Sweet's problem, with uf = 0 and u = uS in the frictionless interior (see Eqs. [4.94] and [4.95]). In contrast, any model for which a = 0 has u = uS + uf in the frictionless interior, since uf = 0 when dw0/dr = 0. One can show that uf > 0 when a > 0 (i.e., when dw0/dr < 0); similarly, one finds that uf < 0 when a < 0 (i.e., when dw0/dr > 0). Since uS > 0, it follows at once that the sum uS + uf is always positive when a > 0 but may change its sign along the radius when a < 0. Therefore, to first order in e, the meridional flow consists of a single cell when a > 0, whereas it may consist of two cells when a < 0. This property is immediately apparent from the fourth column in Table 4.2.

From the solutions presented in Table 4.2, one readily sees that there is a definite intensification of the function u near the surface of models for which a = 0. Obviously, such an intensification does not occur in the limiting case a = 0. Close scrutiny of the second-order corrections indicates that there always exists a surface intensification of the radial component ur = eu\r + e2u2r, no matter whether one has w\ = 0 or w\ = 0. To be specific, in a frictionless model having w0 = 1, one has u1r a 1/p and u2r a 1/p in the surface layers. In contrast, letting w0 = 1 in a frictionless model, one finds that u1r a 1 and u2r a 1/p in these layers. Strictly speaking, then, the case w0 = 1 is mathematically singular since e2|u2r | may become larger than e |u1r | in the surface layers.* Accordingly, a consistent expansion method requires a small amount of differential rotation to O(e1/2), so that one has uir a 1/p in the frictionless solution near the surface. Of course, when turbulent friction is properly taken into account to all orders in the small parameter e, there are no singularities in the components of the

* The case w0 = 1 is the only one for which the function u 1r has no 1 /p singularity in the surface layers. This can happen only if there exists a centrifugal potential that is proportional to r 2[1 — P2(cos 0 )],that is to say, in the case of strict uniform rotation to O(el/2). Note that such a mathematical complication does not occur when the thermally driven currents are caused by disturbing forces other than the centrifugal force of rotation.

circulation velocity u. Figure 4.4 clearly illustrates how the frictional force acts to prevent the appearance of inordinately large radial velocities near the outer boundary. Because of the 1/p term in the frictionless solution that remains valid in the deep interior only, the function ur at first increases toward the surface and then drops rapidly to zero at the free boundary. To be specific, there is an intensification of the radial velocities below the surface, typically by two or three orders of magnitude (see Table 4.2). However, given the extreme smallness of the meridional currents in the bulk of a stellar radiative envelope, the maximum radial speed below the free surface does not exceed 1 cm s-1, which is a far remove from the various evaluations that can be found in the literature.

Figures 4.5 and 4.6 illustrate two second-order solutions for the meridional flow, respectively for a = + 10—3 and a = —10—3 (N = 6, ¡H/xV = 102, and e = 10—4). These curves are quite independent of the parameter e in the deep interior, where the second-order terms make a negligible contribution to the first-order solution. In contrast, it is immediately apparent that, even for a rather low value of e, the second-order terms make a sizeable contribution in the surface layers, where two or even three cells may occur. Note especially the cell in the equatorial belt, when the basic angular velocity decreases with depth (a < 0). Obviously, there is a definite interplay between the meridional flow and the spatial variations of the angular velocity in the surface layers of a stellar radiative envelope. This is quite unfortunate because the actual run of the angular velocity depends on the eddy viscosities, which are poorly known parameters.

Consider again a uniformly rotating, nonmagnetic barotrope. Neglecting viscosity and the inertial terms u ■ grad u in the momentum equation, one readily sees that the velocity u is present only in the equations expressing conservation of mass and energy (Eqs. [4.4] and [4.5]). In the case of a nonspherical star, then, Eqs. (4.3), (4.6), and (3.30) provide four scalar relations among the four functions p, p, T, and V. Indeed, letting p = p0 + epi, etc., and linearizing these equations, one can calculate unequivocally the four nonspherical corrections (i.e., p1, p1, T1, and V1) to a given spherical model. These four corrections are independent of the velocity u. Hence, the potential O is also completely determined, and it does not depend on the velocity u either. Because Eq. (4.47) is derived from Eq. (4.5), which is independent of the remaining equations, there is thus no reason to believe that the constraint (4.47) will be satisfied by the functions (g) and (g—1> that one has derived from the known potential O. Prima facie, this raises serious questions about the validity of Eq. (4.48).

As was pointed out in Section 4.2.2, the Gratton-Mestel proof of the double-cell pattern rests on the fact that their frictionless, nonmagnetic body remains strictly barotropic in spite of the inexorable meridional flow. We may therefore ask the following question: Is it actually possible to obtain such a flow in a uniformly rotating body that has all the properties of a barotrope? Specifically, given the approximations made, Eq. (4.49) implies that one has u ■ grad (r2 sin2 d) = 0, that is, the streamlines of the meridional flow must coincide with the straight lines r sin d = constant (i.e., lines parallel to the rotation axis). This is an impossible requirement, since Eq. (4.38) implies that the streamlines must be closed curves. Moreover, because the meridional velocities in a frictionless system have unwanted singularities at the upper and lower boundaries, there is no reason

Fig. 4.5. Second-order solution for the meridional flow in a Cowling point-source model, with electron-scattering opacity, M = 3M0, N = 6, /^v = 102, e = 10-4, and a = + 10-3. In the inner cell, interior upwelling along the rotation axis is compensated by interior downwelling in the equatorial belt. The sense of circulation is reversed in the outer cell that is adjacent to the rotation axis. Note that there are two cells in the outer layers: One of them is adjacent to the rotation axis, and the other is located in the equatorial belt. Source: Tassoul, M., and Tassoul, J. L., Astrophys. J., 440, 789, 1995.

Fig. 4.6. Same as Figure 4.5, but for a = -10 3. Source: Tassoul, M., and Tassoul, J. L., Astrophys. J.., 440, 789, 1995.

to believe that one can apply the condition n ■ u dS = 0 (4.115)

on each level surface S, since this integral relation implicitly assumes that the velocity u is everywhere finite. We therefore conclude that the Gratton-Mestel result is the consequence of an excessively large number of conflicting assumptions that cannot be met in a realistic model.

Making use of Eq. (4.109), one also has, correct to O(e2),

Ur = eu(r)^(f) + e2 [uo(r) + U2(r)P2P + u4(r)P4(f)1, (4.116)

where u0 = — (1/5)(p12/p)u. A mere comparison with Eq. (4.44) shows that Opik's formula does not provide a reliable solution for the meridional flow in arotating barotrope. It must therefore be disregarded.

Yet, because one has uf a 1/p as r ^ R in Eq. (4.95), it is immediately apparent from Eq. (4.98) that a small amount of differential rotation can have a large effect in the surface layers. Does it imply that meridional velocities of the order of kilometers per second are the rule in the outer layers of an early-type star? The answer to this question is flatly no, because any formula that has a 1 /p singularity cannot possibly satisfy all the basic equations and all the boundary conditions. As a matter of fact, we have shown in this chapter that turbulent friction acting on the meridional flow always prevents huge surface velocities, having |ur | < 1 cm s—1 and \ue | < 102 cm s—1 in the surface layers of a 3M0 star in almost uniform rotation. Obviously, these speeds are much slower than those predicted on the basis of the formulae ur a e/p or ur a e2/p, which are completely inadequate in the outermost surface layers of a rotating star.

Consider a single, nonmagnetic white dwarf that produces its luminosity by cooling of its almost isothermal, degenerate interior. Following closely the analysis given in Sections 4.2.1 and 4.3.1, we shall consider a configuration in slow, almost uniform rotation. Hence, we shall expand about hydrostatic equilibrium in powers of the ratio of centrifugal force to gravity at the equator (see Eq. [4.9]). In spherical polar coordinates (r, l = cos d, the meridional velocity u is

u = e u(r) P2(ß)1r + e v(r)(1 - ß2)—^ 1ß, (4.117)

where, by virtue of Eq. (4.4), v is related to u by the relation

6 pr2 dr

The meridional flow is thus characterized entirely by the radial function u. (Recall that u0 = — ru^f sin d.) We also have

ß ( ) dP1(ß) , ß ( ) dP3(ß) ß1(r )—;-+ ß3(r )

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