where the ak s are numerical coefficients that depend on the effective polytropic index in the surface layers. One can also show that our boundary conditions reduce to u = 0, jxy v' = 0, and (G/p)' = 0, at r = R. These three conditions become y = 0, (4.77)

dx2 dx x and

x ~TT + 2x — - n(n + 2) —- +-----2-y = 0 (4.79)

dx4 dx3 dx2 x dx x2

at x = 0. Finally, since the solution of Eq. (4.76) must join smoothly the frictionless solution at some depth below the free surface, we must also prescribe that y ^ 1, as x (4.80)

Figure 4.2 illustrates the solution of Eq. (4.76) that satisfies conditions (4.77)-(4.80).

1.00

0.75

0.50

0.25

0.75

0.50

0.25

Fig. 4.2. Function y(x) in the surface boundary layer. The frictionless solution, y = 1, is indicated by a dashed line. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49, 317, 1982.

Fig. 4.3. First-order solution for the meridional flow in a Cowling point-source model, with electron-scattering opacity, M = 3M0, and N = 6 in the boundary layers. The streamlines do not penetrate into the convective core, but there is an accumulation of streamlines in the core boundary layer. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J. Suppl., 49, 317, 1982.

Figure 4.3 illustrates the streamlines of the meridional flow in a Cowling point-source model, with electron-scattering opacity, M = 3M0, and N = 6 in the boundary layers (see also columns 2 and 5 in Table 4.2). To this order of approximation, the circulation pattern consists of a single cell (or gyre) extending from the core to the surface, with rising motions at the poles and sinking motions at the equator. Figure 4.3 gives the false

Table 4.2. The first-order velocity field in a 3M0 stellar model (N = 6).

Table 4.2. The first-order velocity field in a 3M0 stellar model (N = 6).

r / R |
a = 0 |
a = +10 |
-3 |
a = -10-3 |
a=0 |
a = +10—3 |
a = —10—3 | ||

0.283182 |
0. |
0. |
0. |
0. |
0. |
0. | |||

0.283200 |
6.2797E- |
5 |
6.2843E- |
5 |
6.2884E- |
5 |
4.8677E- 1 |
4.8657E—1 |
4.8689E—1 |

0.283250 |
2.8603E- |
3 |
2.8590E- |
3 |
2.8609E- |
3 |
5.4638E+0 |
5.4614E+0 |
5.4650E+0 |

0.283300 |
1.2381E- |
2 |
1.2375E- |
2 |
1.2384E- |
2 |
1.2605E+1 |
1.2599E+1 |
1.2607E+1 |

0.283350 |
2.9318E- |
2 |
2.9305E- |
2 |
2.9325E- |
2 |
1.9126E+1 |
1.9117E+1 |
1.9130E+ 1 |

0.283400 |
5.2093E- |
2 |
5.2070E- |
2 |
5.2105E- |
2 |
2.3471E+1 |
2.3461E+1 |
2.3476E+1 |

0.283500 |
1.0408E- |
1 |
1.0404E- |
1 |
1.0410E- |
1 |
2.3853E+1 |
2.3843E+1 |
2.3858E+1 |

0.283750 |
1.6931E- |
1 |
1.6923E- |
1 |
1.6935E- |
1 |
—9.0272E-1 |
—9.0233E—1 |
—9.0292E—1 |

0.284000 |
1.2678E- |
1 |
1.2673E- |
1 |
1.2681E- |
1 |
— 1.0672E+1 |
— 1.0667E+1 |
— 1.0674E+1 |

0.284500 |
6.8959E- |
2 |
6.8929E- |
2 |
6.8975E- |
2 |
— 1.8782E+0 |
— 1.8774E+0 |
— 1.8786E+0 |

0.285000 |
5.1074E- |
2 |
5.1052E- |
2 |
5.1086E- |
2 |
— 1.4116E+0 |
— 1.4110E+0 |
— 1.4119E+0 |

0.286000 |
3.2826E- |
2 |
3.2811E- |
2 |
3.2833E- |
2 |
—5.5661E—1 |
—5.5637E—1 |
—5.5674E—1 |

0.287500 |
2.1380E- |
2 |
2.1371E- |
2 |
2.1385E- |
2 |
—2.3642E—1 |
—2.3631E—1 |
—2.3647E—1 |

0.290000 |
1.3493E- |
2 |
1.3487E- |
2 |
1.3496E- |
2 |
—9.5189E—2 |
—9.5147E—2 |
—9.5210E—2 |

0.300000 |
5.4137E- |
3 |
5.4113E- |
3 |
5.4149E- |
3 |
— 1.5877E—2 |
— 1.5870E—2 |
— 1.5881E—2 |

0.350000 |
1.4145E- |
3 |
1.4138E- |
3 |
1.4149E- |
3 |
— 1.1254E—3 |
— 1.1250E—3 |
— 1.1256E—3 |

0.400000 |
9.5778E- |
4 |
9.5712E- |
4 |
9.5814E- |
4 |
—4.3693E—4 |
—4.3686E—4 |
—4.3689E—4 |

0.450000 |
8.7991E |
-4 |
8.7913E |
-4 |
8.8045E- |
4 |
—2.7395E |
4 |
—2.7406E—4 |
—2.7377E—4 |

0.500000 |
9.5670E- |
-4 |
9.5554E- |
-4 |
9.5760E- |
4 |
—2.3472E |
4 |
—2.3506E—4 |
—2.3431E—4 |

0.550000 |
1.1540E- |
-3 |
1.1521E- |
-3 |
1.1557E- |
3 |
—2.7174E- |
4 |
—2.7246E—4 |
—2.7093E—4 |

0.600000 |
1.4845E- |
-3 |
1.4810E- |
-3 |
1.4876E- |
3 |
—4.1497E- |
4 |
—4.1625E—4 |
—4.1354E—4 |

0.650000 |
1.9827E- |
-3 |
1.9763E- |
-3 |
1.9886E- |
3 |
—7.6285E |
—4 |
—7.6458E—4 |
—7.6085E—4 |

0.700000 |
2.6988E |
-3 |
2.6874E |
-3 |
2.7097E- |
3 |
— 1.5328E- |
—3 |
— 1.5329E—3 |
— 1.5322E—3 |

0.750000 |
3.6954E |
-3 |
3.6767E |
-3 |
3.7134E- |
3 |
—3.1869E- |
—3 |
—3.1738E—3 |
—3.1991E—3 |

0.800000 |
5.0444E |
-3 |
5.0257E- |
-3 |
5.0622E- |
3 |
—6.7455E |
—3 |
—6.6620E—3 |
—6.8274E—3 |

0.850000 |
6.8240E- |
-3 |
6.8950E |
-3 |
6.7522E- |
3 |
— 1.4717E- |
—2 |
— 1.4254E—2 |
— 1.5178E—2 |

0.900000 |
9.1156E- |
-3 |
1.0170E- |
-2 |
8.0606E- |
3 |
—3.4804E |
—2 |
—3.1660E—2 |
—3.7943E—2 |

0.925000 |
1.0479E- |
-2 |
1.4505E- |
-2 |
6.4533E- |
3 |
—5.7473E |
—2 |
—4.7300E—2 |
—6.7641E—2 |

0.950000 |
1.2003E- |
-2 |
3.2423E |
-2 |
-8.4177E- |
3 |
— 1.0585E- |
—1 |
—5.9876E—2 |
— 1.5182E—1 |

0.975000 |
1.3690E |
-2 |
2.4357E |
-1 |
-2.1619E- |
1 |
—2.5801E |
—1 |
—3.3609E—1 |
— 1.7991E—1 |

0.980000 |
1.4105E |
-2 |
5.0824E- |
-1 |
—4.8003E- |
1 |
—3.3131E- |
—1 |
3.8968E+0 |
—4.5594E+0 |

0.985000 |
1.4427E- |
-2 |
1.4525E+0 |
-1.4237E+0 |
—4.8428E |
—1 |
—3.6741E+0 |
2.7055E+0 | ||

0.990000 |
1.2628E- |
-2 |
2.6253E+0 |
—2.6000E+0 |
—7.4853E |
—1 |
— 1.1203E+2 |
1.1054E+2 | ||

0.995000 |
7.2169E- |
-3 |
2.0576E+0 |
—2.0432E+0 |
—9.4255E |
—1 |
—2.5817E+2 |
2.5629E+2 | ||

0.997500 |
3.6952E |
-3 |
1.1078E+0 |
— 1.1005E+0 |
—9.7945E |
—1 |
—2.9149E+2 |
2.8954E+2 | ||

0.999000 |
1.4913E- |
-3 |
4.5329E |
-1 |
—4.5031E- |
1 |
—9.9211E |
—1 |
—3.0098E+2 |
2.9900E+2 |

1.000000 |
0. |
0. |
0. |
—9.9954E |
—1 |
—3.0601E+2 |
3.0401E+2 |

Source: Tassoul, M., and Tassoul, J. L., Astrophys. J., 440, 789, 1995.

Source: Tassoul, M., and Tassoul, J. L., Astrophys. J., 440, 789, 1995.

\o impression that the streamlines penetrate into the core. Actually, they are closed curves, but there is such an accumulation of streamlines in the core boundary layer (Rc < r < Rc + Sc) that a clear depiction is impossible without enlarging this narrow band. As was expected, because we have made allowance for turbulent friction in the radiative envelope, matter is now flowing freely along its upper and lower boundaries. Moreover, there are no mathematical singularities in the meridional flow; the circulation velocities remain uniformly small everywhere in the radiative envelope. This is a definite improvement over Sweet's frictionless solution.

By making use of Eq. (4.28), we can now solve Eq. (4.57) for the function w1. One finds that dP1 dP3

d ix d x where P1(|) = x and 2P3(|) = 5|3 — 3|. The nondimensional functions »1 and »3 are governed by the following inhomogeneous equations:

Since the component arv of the Reynolds stresses must vanish at the free surface, one has

Assuming that the convective core is uniformly rotating with angular velocity Œ0, we shall also let fa(Rc) = 0 and p3(Rc) = 0. (4.85)

Thus, once we have obtained the functions u and v, Eqs. (4.82)-(4.85) can be solved to give a unique solution. The nonuniform rotation rate follows at once from Eq. (4.55).

It is immediately apparent from Eqs. (4.82) and (4.83) that eand e\fi3\ are of the order of (epurhxV), where brackets indicate a suitable mean value. By virtue of Eqs. (4.37) and (4.54), one readily sees that e|wi| ^ tV/tES. To first order in e, then, the convergence of expansion (4.55) implies that e|w1| < 1 or tV < tES. Letting xV = 10w Xrad, one can show that this requirement implies that e106-N < 1 in a 3M0 star. In a typical rotating star having e ^ 10-2, one must thus let N ^ 5-6. This value is quite similar to those encountered in geophysics (see Eqs. [2.66] and [2.67]). If the condition e 106-N < 1 is not met, however, one can no longer make use of expansion (4.55); that is to say, the full nonlinearity of Eq. (4.49) must be retained in the calculations.

For the sake of simplicity, we shall consider a slowly rotating star for which one has tH ^ tES in its radiative envelope. Hence, we can essentially let ^ = i2(r) + &(r, d), with | Q | « Q. In this case, by virtue of Eqs. (4.3)-(4.6) and (4.50)-(4.51), the circulation velocity u can still be represented by Eq. (4.28). Accordingly, if we let

Eq. (4.49) becomes d2 w /4 f dr2 \r f d w 2 p dr 5 fiV

1 dw

2 dr

where we have neglected the contributions to the function Q. At the free surface, we have dw

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