where SNuc is the rate of variation of mean molecular weight caused by nuclear burning.
Following current practice, we shall expand about hydrostatic equilibrium in powers of the small parameter
where is the (constant) overall rotation rate. Hence, we have u = euj + e 2u2 +--------(5.21)
As explained in Section 4.3.1, the truncated expansion (5.22) is asymptotically convergent provided one has tV < tES, where tV is the viscous time and tES is the circulation time (see Eqs. [4.37] and [4.54]). Since we are neglecting the continuous removal of angular momentum from the surface convective layers, we shall assume strict solid-body rotation to 0(e1/2), that is, we shall let wo = 1 in Eq. (5.22). Correct to 0(e), one can write p = po + epi (5.23)
and similar expansions for the pressure and the gravitational potential. Now, because we want to recover a spherically symmetric model in the limit e ^ 0, Eq. (5.18) implies that, correct to 0(e), one must write
and a similar expansion for the temperature. By assumption, we have /¿0 = /¿0(r, t) and T0 = T0(r, t) in the spherical model corresponding to e = 0. Since we want to obtain a solution that possesses full internal consistency, it follows at once from Eqs. (5.19) and (5.21) that the function /¿0 must satisfy the following equation:
d t where S0(r, t) is the (prescribed) rate of variation of the mean molecular weight in the reference spherical model. Here we shall assume that, in spherical polar coordinates (r, 9, y), one initially has p,(r, 9, t = 0) = /¿0(r, 0) = constant, (5.26)
where the values of /¿0 are then allowed to change in time in a manner that depends on the given function S0 and on Eq. (5.25).
In Section 4.2.1 we have shown that the functions p1, p1, and V1 can be obtained from Poisson's equation and the poloidal part of the momentum equation, which do not depend on px1 and T1. In particular, the continuity of gravity across the outer nonspherical surface implies that p1 = p1fi(r, t) + pu(r, t)P2(cos 9) (5.27)
and similar expansions for p1 and V1, where P2(cos 9) is the Legendre polynomial of degree two. By virtue of Eq. (5.18), however, the expansions for fz1 and T1 contain, in principle, an infinite number of additive terms of the form /¿1j2k(r, t)P2k(cos 9) and T1j2k(r, t)P2k(cos 9), with k = 0, 1, 2, If so, then, the radial component u 1r should also contain an infinite number of additive terms of the form u12k(r, t)P2k(cos 9), with k = 0, 1, 2, Obviously, these terms essentially depend on the initial /¿-distribution in the nonspherical model.
Fortunately, by making use ofEq. (5.26), which is a most plausible initial condition, one can easily show that all terms belonging to k = 2, 3, 4,... must identically vanish from ¿1, T1, and u1r. Indeed, since p12k = p1>2k = 0 when k > 2, it follows from
Eq. (5.18) that one has T1>2k/ T0 = /¿1j2k//¿0 (= a2k, say) for these values of k. Hence, for each k (> 2) Eq. (5.16) implies that ul 2k is a linear and homogeneous function of a2k and its derivatives. Next, linearizing Eq. (5.19) and eliminating ul 2k, one obtains a homogeneous differential equation for each a2k, when k > 2. Now, it readily follows from Eq. (5.26) that a2k(r, 0) = 0 since, by assumption, our initial model is chemically homogeneous. One can also let a2k(0, t) = a2k(Rn, t) = 0 for all t (> 0), where Rn is the radius of the sphere outside which (at the prescribed level of numerical accuracy) nuclear burning and the /¿-gradient may be neglected. Since for each k (> 2) the function a2k is the solution of a linear and homogeneous differential equation, these initial and homogeneous boundary conditions imply that one has a2k (r, t) = 0 for all t (> 0), when k = 2, 3, 4,
In other words, starting from an initially homogeneous core, we can rightfully write i¿i = /¿i,0(r, t) + /¿i,2(r, t)P2(cos 6) (5.28)
and a similar expansion for Tl. The corresponding meridional velocity is, therefore, dP2(cos 6)
ui = u(r, t)P2(cos6)lr - rv(r, t)sin6—p-- 1e. (5.29)
d cos 6
Equation (5.2) provides the link between the functions u and v. One finds that
6 pr2 dr where we have omitted the subscript "0" from the density in the spherical model.
Correct to O(e3/2), the back reaction of the first-order part of the meridional flow on the constant overall rotation is dPi(cos 6) dP3(cos 6)
The functions j31 and satisfy two equations that are quite similar to Eqs. (4.82) and (4.83), with 3^1/31 and dfí3/81 being retained since u and v depend on time.
Now, it is immediately apparent that the functions p12 and p12 can be obtained from Eqs. (4.24) and (4.25). By virtue of Eq. (5.18), however, Eq. (4.27) must be replaced by
T \p' pj where the function h can be obtained from Eq. (4.23). For shortness, we have also let a = . (5.33)
As usual, we have omitted the subscript "0" from the functions in the spherical model corresponding to e = 0. A prime denotes a derivative with respect to the radial variable r.
Inserting next these solutions into the energy equation, one finds that the radial function u can be written in the form u = ua(r, t) + u¿[a(r, t), r, t], (5.34)
thus indicating that the large-scale meridional flow is the sum of "^-currents" and "f-currents". After collecting and rearranging terms, we obtain
where the functions a0(r, t) and a1(r, t) depend on the reference spherical model, and n is the effective polytropic index (see Eq. [4.33]). The function l is the net amount of energy crossing the spherical surface of radius r per second, that is, l = —4n r 2x T'. (5.36)
Parenthetically note that Eq. (5.35) merely reduces to Sweet's function (4.32) in the outer parts of the Sun's radiative core, where one has l = L, eNuc = 0, and fx = constant. Similarly, one can show that the function u ^ has the form l n + 1 T „
where D"a is a second-order differential operator acting on the function a, that is,
dr2 dr where the functions A0(r, t) and A1 (r, t) depend on the reference spherical model. Equations (5.37) and (5.38) were originally obtained by Mestel (1953). Making use of Eqs. (5.19) and (5.25), one also has d
d t where 5i>2 depends on the choice that is made for the function 5Nuc. Substituting for u in accordance with Eq. (5.34), one can calculate the function a from Eq. (5.39), which is parabolic in structure. Thence, the radial function u can be obtained from Eqs. (5.34)-(5.38).
Now, one readily sees that n ^ 3/2 near the top of the radiative core, thus implying the existence of a mathematical singularity in our frictionless solution. As explained in Section 4.3.1, this major inadequacy can be resolved by making use of the thermo-viscous boundary-layer solution depicted in Figure 4.1, letting x = (Rc — r)/8c in Eq. (4.70) since we are now approaching the singularity from below the inner boundary. This modification is not essential for the subsequent discussion, however, because the interaction between the fx-distribution and the rotationally driven currents takes place in the bulk of the Sun's radiative core, away from the core-envelope interface.
Numerical calculations have been performed by making use of an evolutionary sequence of a standard 1 M0 model. Figures 5.3 and 5.4 illustrate at selected instants the functions u and a in the chemically homogeneous part of the Sun's radiative core (see Eqs. [5.34] and [5.35]). It is worth noting that the function a always remains much smaller than unity. To order e, it is thus correct to make use of the truncated expansion (5.24) and the linearized equation (5.39).
It is immediately apparent from Figure 5.3 that, almost from the start, the fx-currents oppose the ^-currents - the large-scale circulatory motions die out as the fx-gradient
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