In Section 3.3.1 we noted that the conditions of mechanical and radiative equilibrium are, in general, incompatible in a rotating barotrope. This paradox can be solved in two different ways: Either one makes allowance for a slight departure from barotropy and chooses the angular velocity ^ = Q.(rn, z) so that strict radiative equilibrium prevails at every point or one makes allowance for large-scale motions in meridian planes passing through the rotation axis. The first alternative is mainly of academic interest because there is no reason to expect rotating stars to select zero-circulation configurations. Moreover, these baroclinic models are thermally unstable with respect to axisymmetric motions, as well as dynamically unstable with respect to nonaxisymmetric motions (see Sections 3.4 and 3.5). Hence, the slightest disturbance will generate three-dimensional motions and, as a result, a large-scale meridional circulation will commence. The second alternative was independently suggested by Vogt (1925) and Eddington (1925), who pointed out that the breakdown of strict radiative equilibrium in a barotrope tends to set up slight rises in temperature and pressure over some areas of any given level surface and slight falls over other areas. The ensuing pressure gradient between the poles and the equator thereby causes a flow of matter. In fact, it is the small departures from spherical symmetry in a rotating star that lead to unequal heating along the polar and equatorial radii, which in turn causes large-scale currents in meridian planes. Slow but inexorable, thermally driven currents also exist in a tidally distorted star, as well as in a magnetic star, since the tidal interaction with a companion and the Lorentz force both generate small departures from spherical symmetry in a star. Obviously, it is the causal relation between nonsphericity and meridional circulation that makes the stellar problem entirely different from those expounded in Sections 2.5 and 2.6. This fact strongly suggests that well-known results obtained in geophysics (such as geostrophy and Ekman layers) should not be applied indiscriminately to a stellar radiative zone. I shall comment further on these important matters in Section 4.8.
In Section 4.2.1 we will obtain the steady circulation pattern in the radiative envelope of a uniformly rotating, frictionless star. Following Sweet (1950), we shall thus calculate the meridional flow generated by the nonsphericity of a chemically homogeneous region in slow uniform rotation. Section 4.2.2 presents a critical reassessment of his solution, which becomes infinite both at the free surface and at the core-envelope interface, and which also fails to take into account the transport of specific angular momentum by the meridional flow. In Sections 4.3 and 4.4, by making use of the eddy-mean flow interaction, which takes place continuously in a stellar radiative envelope, we obtain a simple but adequate description of the mean state of motion in a rotating star. This solution, which is free of the objections that can be made about Sweet's frictionless solution, satisfies all the boundary conditions and all the basic equations. Thermally driven currents in cooling white dwarfs are considered next in Section 4.5. Section 4.6 is devoted to the circulatory currents in the radiative envelope of an early-type star, which is a detached component of a close binary, and whose surface is nonuniformly heated by the radiation of its companion. Meridional flows in magnetic stars are considered further in Section 4.7. We conclude the chapter with a general overview of the problem, pointing out the differences and similarities between the large-scale currents that are encountered in geophysics and astrophysics.
Consider a single, nonmagnetic star that has a fully convective core, in which hydrogen burning is taking place, and a chemically homogeneous radiative envelope. Assume also that the axially symmetric star is slowly rotating with the constant angular velocity Œ0. We shall also neglect viscosity and the inertia of the circulation itself. Then, in an inertial frame of reference, the equations governing steady motions in the radiative envelope are d p d V ,
ff where p is the pressure, p is the density, V is the inner gravitational potential, u is the two-dimensional circulation velocity, T is the temperature, and R is the perfect gas constant. For electron-scattering opacity, the coefficient of radiative conductivity has the form
3 Kp where k is a constant. For a simple ideal gas, we also have p
where cV is the specific heat at constant volume.
4.2.1 Sweet's meridional circulation
Since it is the lack of spherical symmetry that causes the meridional flow, we shall first derive from Eqs. (4.1)-(4.3) an expression for the distortion of the level surfaces due to the slow but uniform rotation. Following Milne (1923) and Chandrasekhar (1933), we shall expand about hydrostatic equilibrium in powers of the nondimensional parameter
where M is the total mass and R is the equatorial radius. (In a realistic main-sequence model, e does not exceed the critical value ec ^ 0.4, at which point equatorial breakup is likely to occur.) Letting P2(i) denote the Legendre polynomial of degree two, we thus write, in spherical polar coordinates (r, i = cos d, p(r, i) = po(r) + e[pi,o(r) + pu(r)P2I (4.10)
and a similar truncated expansion for the density. The inner gravitational potential is
V(r, i) = Vo(r) - GM + e[V1,o(r) + Ci,o + Vu(r)P2I)], (4.11) whereas the potential that is appropriate to the surrounding vacuum has the form
where B0 and B2 are constants.
Now, by making use of Eqs. (4.1)-(4.3), one can easily show that the nonradial functions (i.e., p1>2, p1>2, and F12) satisfy the following set of equations:
d2V12 2 dV12 6
dr2 r dr r2
where = GM/R3. A prime denotes a derivative with respect to r. Without confusion, we have omitted the subscript "0" from the functions p0 and p0 that define the (known) model corresponding to e = 0. The continuity of the gravitational field across the free surface, which is a slightly oblate surface, further implies that
We shall not write down the relations between the radial functions (i.e., p10, p10, and V10) since, to first order in e, they are not relevant to the circulation problem. To solve Eqs. (4.13)-(4.15), we shall let
where is a constant. Thence, it is a simple matter to show that the function $2 satisfies the following equation:
with $2(0) = $2(0) = 0. Boundary conditions (4.16) now become
96 Meridional circulation and
Solving for A2 and B2, one obtains
thus ensuring that the inner potential (4.11) smoothly joins the outer potential (4.12). Thus, by letting
we have shown that
where the function can be obtained from Eq. (4.18).
Following Sweet (1950), we now turn to Eqs. (4.4)-(4.6). By making use of the equation of state, one readily sees that the temperature can be expanded as was done for the pressure and density. Hence, we can write
T p p where we have also omitted the subscript "0" from the temperature in the spherical model. Combining Eqs. (4.24)-(4.26), one finds that
If we now make use of Eqs. (4.4) and (4.5), it is a simple matter to show that, correct to first order in e, the circulation velocity has the form u = e u(r) P2(^) 1r + e v(r) (1 - p2) —^ (4.28)
d p, where
6 pr2 dr
(As usual, we also have u0 = -ru^f sin d.) By virtue of Eq. (4.29), the meridional circulation depends on the single function u only.
Inserting our truncated expansions into Eq. (4.5), one finds that d dr v = - -^—(pr 2u). (4.29)
where L is the total luminosity of the model corresponding to e = 0. In establishing this equation, we have made use of the fact that
where x is the coefficient of radiative conductivity in the spherical model. Inserting next solutions (4.25) and (4.27) into Eq. (4.30), we obtain an algebraic equation for the function u. Its solution, u = uS (say), has the form
rm where m is the mass contained within the sphere of radius r, m' = 4npr2, and
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