P T

is the effective polytropic index. This is Sweet's (1950) solution for the meridional flow in the radiative envelope of a star in slow uniform rotation. Equation (4.32) can also be written in the form

2 Lr4

rm h where d ln T (d ln T\ V = —- and Vad = —- .

Combining Eqs. (4.23) and (4.34), one readily sees that \uS| ^ LR2/GM2 in the bulk of a stellar radiative zone. Hence, we have

LR2 eR

where tKH is the Kelvin-Helmholtz time and e is the ratio of centrifugal force to gravity at the equator (see Eq. [4.9]). This result implies at once that the time scale of the meridional flow in the bulk of a radiative envelope (tES, say) is tKH

which is known as the Eddington-Sweet time.

4.2.2 The classical objections

Table 4.1 gives a detailed solution for a Cowling point-source model with electron-scattering opacity. * Here we have Tc = 1.76 06 x107 ¡i MM/R and pc = 4.0779 x 1017 M / R , where ¡i is the mean molecular weight and the remaining barred quantities n

* This simple numerical model, with power-law opacity and point-source energy generation, was originally discussed by Thomas George Cowling (1906-1990) in 1935. It consists of a convective core that contains all the energy sources and a radiative envelope. See, e.g., Cox, J. P., and Giuli, R. T., Principles of Stellar Structure, Sections 19.2a and 23.4, New York: Gordon and Breach, 1968; Tayler, R. J., Quart. J. R. Astron. Soc., 32, 201, 1991.

Table 4.1. Physical properties of a Cowling point-source model.

r

m

T

P

n

h

__us _

r vs

( R)

(M )

(Tc )

( Pc)

( M / R)

( L R2/ M2)

( L R2/M2)

0.00000

0.

1.0000E+0

1.0000E+00

1.5000

0.

0.05000

2.4440E-

3

9.8719E-

1

9.6829E-

01

1.5000

5.8690E+12

0.10000

1.8886E-

2

9.4965E-

1

8.7883E-

01

1.5000

2.2905E+13

0.15000

6.0178E-

2

8.8988E-

1

7.4702E-

01

1.5000

4.9487E+13

0.20000

1.3169E-

1

8.1176E-

1

5.9370E-

01

1.5000

8.3212E+13

0.25000

2.3234E-

1

7.2002E-

1

4.3991E-

01

1.5000

1.2126E+14

0.28318

3.1197E-

1

6.5413E-

1

3.4607E-

01

1.5000

1.4751E+14

infinite

infinite

0.28319

3.1199E-

1

6.5412E-

1

3.4605E-

01

1.5001

1.4752E+14

3.7196E-

1

-2.2226E+3

0.28320

3.1201E-

1

6.5409E-

1

3.4602E-

01

1.5002

1.4752E+14

1.6414E-

1

-4.3284E+2

0.28400

3.1403E-

1

6.5247E-

1

3.4388E-

01

1.5070

1.4816E+14

3.5870E-

3

-2.0748E-1

0.29000

3.2927E-

1

6.4042E-

1

3.2802E-

01

1.5577

1.5291E+14

4.2638E-

4

-3.0082E-3

0.30000

3.5507E-

1

6.2072E-

1

3.0244E-

01

1.6400

1.6081E+14

1.7107E-

4

-5.0179E-4

0.35000

4.8749E-

1

5.2904E-

1

1.9237E-

01

2.0069

1.9897E+14

4.4700E-

5

-3.5569E-5

0.40000

6.1480E-

1

4.4806E-

1

1.1381E-

01

2.2980

2.3356E+14

3.0266E-

5

-1.3809E-5

0.45000

7.2584E-1

3.7703E-

1

6.3121E-

02

0.50000

8.1507E-1

3.1511E-

1

3.3055E-

02

0.55000

8.8192E-1

2.6136E-

1

1.6412E-

02

0.60000

9.2897E-1

2.1478E-

1

7.7220E-

03

0.65000

9.6012E-1

1.7436E-

1

3.4200E-

03

0.70000

9.7944E-1

1.3919E-

1

1.4046E-

03

0.75000

9.9054E-1

1.0842E-

1

5.2052E-

04

0.80000

9.9631E-1

8.1376E-

2

1.6572E-

04

0.85000

9.9888E-1

5.7460E-

2

4.1255E-

05

0.90000

9.9979E-1

3.6182E-

2

6.4894E-

06

0.95000

9.9999E-1

1.7139E-

2

3.2677E-

07

0.99000

1.0000E+0

3.2894E-

3

4.4332E-

10

0.99900

1.0000E+0

3.2597E-

4

4.2756E-

14

0.99990

1.0000E+0

3.2568E-

5

4.2602E-

18

0.99999

1.0000E+0

3.2565E-

6

4.2587E-

22

1.00000

1.0000E+0

0.

0.

2.5195

2.6420E+14

2.7805E

-5

-8.6574E-

6

2.6823

2.9180E+14

3.0232E

-5

-7.4173E-

6

2.7982

3.1793E+14

3.6467E

-5

-8.5870E-

6

2.8779

3.4421E+14

4.6910E

-5

-1.3113E-

5

2.9306

3.7207E+14

6.2653E

-5

-2.4106E-

5

2.9636

4.0262E+14

8.5282E

-5

-4.8436E-

5

2.9829

4.3665E+14

1.1677E-

-4

- 1.0071E-

4

2.9932

4.7471E+14

1.5940E-

-4

-2.1316E-

4

2.9979

5.1714E+14

2.1564E-

-4

-4.6505E-

4

2.9996

5.6410E+14

2.8805E

-4

- 1.0998E-

3

3.0000

6.1562E+14

3.7930E

-4

-3.3448E-

3

3.0000

6.6006E+14

4.6775E

-4

-2.2836E-

2

3.0000

6.7044E+14

4.8973E

-4

-2.4429E-

1

3.0000

6.7149E+14

4.9198E-

-4

-2.4593E+0

3.0000

6.7160E+14

4.9220E

-4

-2.4609E+1

3.0000

6.7161E+14

4.9222E-

-4

infinite

Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49, 317, 1982.

are expressed in solar units instead of in cgs units. The sixth column must be multiplied by M/R to obtain the values of the function h in cgs units. Similarly, once the last two

columns have been multiplied by L R /M , they provide Sweet's solution - uS and r vS - in cgs units. His solution for the meridional flow consists of a single cell, with interior upwelling at the poles and interior downwelling at the equator (see Figure 4.3). Unfortunately, as was expected from Eqs. (4.29) and (4.32), one finds that uS a 1/(n — 3/2) and vS a 1/(n — 3/2)2 at the core boundary, whereas uS = 0 and vS a p'/p at the free surface. This implies at once that the frictionless solution does not stream along the boundaries. To be specific, without mass loss, a consistent solution of the problem must be such that n ■ u = 0, with |u| finite, (4.38)

at the boundary r = R (see Eq. [2.20]). A similar condition applies at the core boundary r = Rc if we assume that the circulatory currents do not penetrate into the convective region. Yet, one finds that ur a 1 and ue a (R — r)—1, (4.39)

near the free surface, and ur a (r — Rc)—1 and ue a (r — Rc)-2, (4.40)

near the core-envelope interface.

As was shown by Baker and Kippenhahn (1959), the situation is even worse when the prescribed rotation law is nonuniform. In that case, neglecting viscous friction and the inertial terms u ■ grad u, they found that Eq. (4.36) must be replaced by

where a0 and j30 are constants of order unity, p is the mean density, and AO is a measure of the prescribed nonuniform rotation rate. Hence, for electron-scattering opacity, Eq. (4.39) must be replaced by ur a (R — r)—3 and ug a (R — r)—3, (4.42)

near the free surface. As they noted, in radiative regions near the surface of a differentially rotating star one can thus expect much higher meridional velocities than are calculated on the assumption of strict uniform rotation. This matter will be considered further in Section 4.4.1.

From the viewpoint of astronomy, Eqs. (4.36) and (4.41) are quite satisfactory, since they provide an order of magnitude of the circulation velocities in the bulk of a radiative envelope. They also point to an apparent difference between solid-body rotation and differential rotation, the latter causing a definite intensification of the meridional currents in the surface layers of an early-type star. Unfortunately, these two formulae are not directly applicable in the surface regions, because none of them satisfies the kinematic boundary condition (4.38) at the outer boundary. Moreover, one readily sees that the 1/p singularity in Eq. (4.41) implies that one has |pu ■ gradu| a 1/p, thus invalidating the method of solution in the surface layers. Note also that in both solutions one has neglected the inexorable transport of angular momentum by the meridional currents.

Another serious objection was raised by Opik (1951), who noted that Sweet's solution for the radial component of the circulation velocity,

should be replaced by

2n Gp J

If so, then, the meridional flow consists of two distinct cells (or gyres, as they say in geophysics) separated by the level surface with density p = p* (say) given by ^2 = 2n Gp*. The following analytical proof of this property was broached by Gratton (1945) and Mestel (1966). Consider a chemically homogeneous radiative envelope in uniform rotation. Neglect friction and the inertia of the meridional currents. Then, by making use of Eqs. (3.31)-(3.36), one can rewrite Eq. (4.5) in the form pA(O) u ■ grad O = -f (O) (4nGp - ) - f'(O)g2, (4.45)

where

and g = dO/dn is the magnitude of the effective gravity. (Remember that g varies over a level surface!) Dividing Eq. (4.45) by g and integrating over a level surface, we obtain f (O) (4nGp - 2^0) <g-1} + f (O)(g) = 0, (4.47)

since in a steady state there can be no flux of matter across a level surface. (Angular brackets designate a mean value over a level surface.) From Eqs. (4.45) and (4.47), it is clear that one has

If the function f (O) vanishes for a value O = O* (say), this equation implies that the meridional currents do not cross the corresponding level surface. By virtue of Eq. (4.47), one has f (O) = 0 on the level surface with density p*(O*) given by ^2 = 2nGp*. This concludes the analytical proof that there apparently exists a double-cell pattern in a uniformly rotating radiative envelope.

As we shall see in Section 4.4.1, the Gratton-Mestel proof of the double-cell pattern is incorrect; Opik's equation (4.44) is also quite inadequate for describing the meridional flow in a radiative envelope.

4.3 A consistent first-order solution

In Sections 2.5.1 and 2.6.2 we have presented frictionless solutions that describe large-scale flows in the Earth's atmosphere and in the oceans (see Eqs. [2.79] and [2.113]). In both cases, however, these solutions fail to satisfy the appropriate boundary conditions. This is the reason why turbulent friction had to be retained in narrow layers near the natural boundaries (see Eqs. [2.87]-[2.88] and [2.119]). The importance of eddy viscosity near the boundaries is directly related to the fact that the viscous force contains second-order derivatives in the velocities (see Eq. [2.65]). Hence, if eddy viscosity is neglected altogether in Eq. (2.64), the order of this equation is reduced so that its solutions can no longer satisfy all the boundary conditions that are required by the nature of the problem. As we know, the only way to satisfy all these conditions is to retain turbulent friction in thin boundary layers, where the velocities may vary rapidly in space. Then, the frictional force will be of the same order as the nonfrictional terms, notwithstanding the smallness of the coefficients of eddy viscosity. This is the key idea involved in boundary-layer theory. Not unexpectedly, a boundary-layer analysis of the thermally driven currents in the radiative envelope of a nonspherical star is a much more complex problem because it involves both the momentum equation and the energy equation. This will become apparent in the following pages.

In Section 4.2 we calculated the thermally driven currents in a stellar radiative envelope that we compel to rotate as a solid body. To obtain a fully consistent solution in a nonmagnetic star, we shall retain turbulent friction in Eqs. (3.125), (3.126), and (3.133). Hence, it is no longer necessary to prescribe the rotation rate, since the transport of angular momentum by the meridional flow can now be adjusted steadily so as to balance the effects of friction on the angular velocity. By virtue of Eq. (3.133), neglecting the XV effect, we thus have sin26 d ( 4 dQN 1 d ( 3 dQN

Similarly, we shall replace Eqs. (4.1) and (4.2) by the following equations:

d6d6 v y where Fr and F6 are the poloidal components of the turbulent viscous force per unit volume (see Eqs. [3.125] and [3.126]). Equations (4.3)-(4.6) remain unaffected by eddy viscosity. Equations (4.49)-(4.51) and (4.3)-(4.6) thus provide seven relations among the seven unknown functions Q, u, p, p, T, and V.

Because the angular velocity is in general a function of both r and 6, let us write

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