where k2 = k2m + k2z. For a simple ideal gas, we have
Accordingly, we can write grad S = c and
T //J p where we made use of the fact that
Since the rate of diffusion of chemical species is comparable to the (negligible) viscous diffusion rate, we shall also assume that
It follows that
By making use of Eqs. (3.108)-(3.112), we can thus rewrite Eq. (3.106) in the form
1 + €~) SsP = S (a ■ *) - ^ fa ■ ^ , (3.113) n/ p n \ L J
where c = x k2/pcp (3.114) and ^ = grad S/cp, with grad S being defined in Eq. (3.108).
It is now a simple matter to eliminate Sp/p between Eqs. (3.104) and (3.113) to obtain n3 + en2 + An + e B = 0, (3.115)
A = (a ■ $)(a ■ $0) + (a ■ ¥)(a ■ ¥0) (3.116)
B = (a ■ $)(a ■ $0) - (a ■ (a ■ ¥0). (3.117)
In the limiting case e = 0, Eq. (3.115) provides the requisite dispersion relation for discussing dynamical stability (see Section 3.4.2). When e = 0 and A > 0, its three roots are n = ±iVA, which describe stable oscillations, and the trivial root n = 0. When e > 0, the roots can be written in the forms n = ±i a + a and n = b, (3.118)
where a, a, and b are real numbers (see Eqs. [3.102]). According to the Routh-Hurwitz criterion,* a and b are negative if and only if
These two inequalities can be rewritten in the form
(a ■ $)(a ■ $0) - (a ■ ^^d^J (a • ^0) > 0 (3.120)
(a ■ ¥)(a ■ ¥0) + (a ■ (a • ^0) > 0. (3.121)
These are the conditions for thermal stability with respect to axisymmetric motions, when both radiative conductivity and a gradient of chemical composition are taken into account.
Consider first the chemically homogeneous part of a stellar radiative zone. By virtue of Eq. (3.120), thermal instability occurs whenever a vector a can be found that will make (a ■ $)(a ■ $0) negative (i.e., b > 0 in Eq. [3.118]). Figure 3.3 illustrates the case of a dynamically stable barocline (as illustrated in Figure 3.1). It is a simple matter to see that all vectors a that lie in the cross-hatched region make the body thermally unstable at that point. Obviously, the only way to prevent this instability in a star is to remove the cross-hatched region at every point. This can be done only if the vector $ points in the m direction, that is, if
d z d m at every point of the radiative interior. This result was originally obtained by Goldreich and Schubert (1967) and, independently, by Fricke (1968).
* See, e.g., Handbook of Applied Mathematics (Pearson, C. E., ed.), p. 929, New-York: Van Nostrand, 1974.
Following Shibahashi (1980), let us consider the implications of condition (3.121) in a chemically homogeneous region. In that case, if one can find a vector a that will make (a ■ ^)(a ■ ^0) negative, thermal overstability occurs in the system (i.e., a > 0 in Eq. [3.118]). Figure 3.4 clearly shows that all vectors a lying in the cross-hatched region generate overstable motions at that point. This oscillatory instability can be removed only if the vectors ^ and point to the same direction. By virtue of Eq. (3.88), this requirement also implies that the vector $ points in the m direction. Again, this is true only if condition (3.122) is satisfied at every point of the radiative interior.
Now, is it possible to maintain the chemically inhomogeneous part of a stellar radiative zone in static equilibrium with the steady rotation law ^ = Q.(m, z)? Condition (3.120) shows that a stable gradient of chemical composition (i.e., grad / < 0) often has a stabilizing influence on all unstable motions in the wedge between the surfaces m = constant and j = constant. Accordingly, a suitable stratification of mean molecular weight might well prevent the Goldreich-Schubert-Fricke instability from occurring in a baroclinic star. However, by making use of Eqs. (3.108) and (3.121), one also sees that Shibahashi's oscillatory instability is probably little affected by a stable /¿-gradient. This is a mere consequence of the fact that the overstable motions are located in the wedge between the surfaces p = constant and S = constant, which differ little from the surfaces / = constant. Obviously, further discussion of the effects of a /¿-gradient in a baroclinic star necessarily requires the use of a particular model for the radiative interior.
To conclude, let us note that these thermal instabilities also are a form of baroclinic instability, since both of them are driven by the baroclinicity of the basic state. However, they differ from the baroclinic instability of the kind discussed in Sections 2.7.2 and 3.4.3 in two obvious ways. First, unlike the usual baroclinic instability, which is associated with nonaxisymmetric motions, they are axisymmetric instabilities. Second, because they depend upon the relaxation of the isentropic constraint, their time scale is certainly much longer than the time scale for the usual baroclinic instability. At this writing, the time scale for angular momentum transport by these thermal instabilities remains controversial, ranging in the literature from the Kelvin-Helmholtz time to the Eddington-Sweet time of large-scale meridional currents (see Eq. [4.37]). This is probably of no great consequence, however, because the dynamical instabilities with respect to nonaxisymmetric disturbances will generally dominate in a rotating star.
The first step toward understanding the dynamics of a rotating star requires that we simplify the basic equations so that they describe only the largest scale of motion. As was pointed out in Section 2.4, however, large-scale flows do not exist in isolation in a huge natural system, such as a star. This is because ever-present nonaxisymmetric instabilities in a rotating star generate a wide spectrum of eddylike motions.* These small-scale disturbances give rise, by nonlinear processes, to fluxes of heat and momentum and, hence, influence the dynamics of the largest scale motions. In geophysics, this is called the eddy-mean flow interaction. This global approach rests essentially on a dynamical linkage between the ever-present eddylike motions (which we call "anisotropic turbulence" because effective gravity and rotation define two preferential directions in a star) and the mean flow (that is, the overall rotation and concomitant motions in meridian planes passing through the rotation axis). As usual, the role of these eddylike motions is simply parameterized in frictional form through the use of eddy viscosities and related coefficients. Not unexpectedly, these coefficients attain much larger values in a convectively unstable region than in a stellar radiative interior.
Neglecting molecular viscosity and omitting the overbars, we can thus rewrite Eq. (2.58) in the form
Dt p p where F is the turbulent viscous force per unit volume, which can be written as the vectorial divergence of Reynolds stresses (see Eq. [2.59]). In spherical polar coordinates (r, d, the mean velocity v is v = ur lr + ue 1e + Q,r sind 19, (3.124)
where ur and u 6 are the components of the two-dimensional meridional velocity u.
* As was noted by Balbus and Hawley (1998) and others, small-scale magneto-rotational instabilities play an important role in generating turbulence in accretion disks. Under very specific circumstances, similar instabilities might be relevant to the study of turbulent motions in stellar radiative zones.
For axisymmetric motions, the poloidal part of Eq. (3.123) has the components d ur d ur Ug dur d V d p pi — + ur — + — — - ^ ) = - p—--— + pQ2r sin2 0
r sin 0
sin 0d 2 d
They depend on the Reynolds stresses orr, o00, oww, and or0. These quantities can be simply expressed as d ur
ur u0 cot0
where fV and fiH are the vertical and horizontal coefficients of eddy viscosity. Equations (3.125) and (3.126) thus depend on two parameters. Of course, this can only be a very crude model, but it does make allowance for a difference in momentum transfer between the vertical (i.e., along the effective gravity) and horizontal directions.
For axisymmetric motions, the toroidal part of Eq. (3.123) depends on the Reynolds stresses arv and o0q). Following Rudiger (1980) and others, we shall let and dQ .
These relations depend on the eddy viscosities and two additional parameters, XV and XH, which represent the influence of global rotation on anisotropic turbulence. The free parameter XH identically vanishes whenever the eddylike motions have horizontal symmetry, being then isotropic in planes perpendicular to the effective gravity. To a good degree of approximation, the XH term can be neglected in a slowly rotating star. In that
case, by making use of Eqs. (3.131) and (3.132), one can show that the y component of Eq. (3.123) has the form dQ (dQ Q\ u0 (dQ
p — + p ur[ — + 2- + p—[ — + 2Q cot 0 dt \Br r J r \B0
1 d ( 4 dQ 3 \ 1 1 ( . 3 9Q\ = W4 — + Vr3Q + - Ph sin3 0 — ), (3.133)
which depends on three independent parameters only. As we shall see in Section 5.2.1, however, the kH term makes a nonvanishing contribution to the toroidal viscous force acting in the solar convective envelope. In that sense, thus, the Sun is not a slowly rotating star.
Now, as was originally pointed out by Schatzman (1969), anisotropic turbulence generated by the nonaxisymmetric instabilities may contribute to the diffusion of chemical elements within a stellar radiative zone. More recently, Press (1981) suggested that internal waves generated by chaotic motions at the boundary of a convective zone might also lead to species mixing in stably stratified regions. As usual, lacking any better description of all these eddy and/or wave events, we shall lay emphasis on the mean properties, using gross parameterizations of the smallest scale motions. For axisymmetric motions, the turbulent transport of a chemical element with concentration c can be described by the following equation:
where p is the density and DV and DH are the vertical and horizontal coefficients of eddy diffusivity. (D/Dt is the total derivative.) As was noted by Fujimoto (1988) and others, however, vertical mixing is probably much less efficient than horizontal mixing, especially in a strongly stratified system. Indeed, for element mixing, work has to be done against gravity, so that the vertical displacements may be easily inhibited by the buoyancy force. In contrast, the instabilities responsible for horizontal turbulence are the barotropic and baroclinic instabilities, which are caused by latitudinal variations of angular velocity and temperature along the isobaric surfaces (see Section 3.4.3). Recall that these instabilities are operative for all positive values of the Richardson number Ri whereas the usual shear-flow instability, which is associated with a vertical shear in the rotational motion, is operative only when condition (3.101) is satisfied.
Various measurements in the laboratory and in the Earth's atmosphere indicate that, under stable conditions, the eddy diffusivities of matter and momentum decrease with increasing stability. These studies also show that the turbulent diffusion of matter is a much less effective process than the turbulent diffusion of momentum in a stably stratified system. Specifically, it is found that the ratio of eddy diffusivity to eddy viscosity, p DV/pV ,is of the order of a few tenths for Ri < 1, whereas for Ri > 1 this ratio steadily decreases to zero as Ri ^ <x> (e.g., Turner 1973). These results are quite interesting because they strongly suggest that the ratio pDV/pV can also be assumed to be much smaller than one in a stellar radiative interior. This matter will be discussed further in Section 5.4.1.
In this section we have developed a theoretical framework that describes the largest scale of motion in a rotating star. In particular, whereas the poloidal part of the momentum equation depends on two independent parameters (i.e., ¡V and ¡¡H), it is found that its toroidal part depends on at least three independent parameters (i.e., ¡¡V, ¡¡H, and XV). As was seen in Section 3.3, the parameter XV is of paramount importance because it prevents solid-body rotation in a convective envelope. Note also that the equation governing turbulent diffusion of matter in an axially symmetric star depends on two additional parameters (i.e., DV and DH). Equations (3.127)—(3.132) specify the Reynolds stresses in such away that Eqs. (3.125), (3.126), and (3.133) represent a closed set of equations for the large-scale flow. (Compare with Eqs. [2.60]-[2.65].) Unfortunately, because there is no a priori justification for this particular model, it must be borne in mind that the eddy-viscosity coefficients cannot be calculated from first principles alone. A similar remark can be made about the eddy-diffusivity coefficients, DV and DH, since the ad hoc nature of the underlying model precludes a deterministic calculation of their values in a rotating star.
As was noted in Section 2.4, measurements in the Earth's atmosphere and in the oceans show that the eddy viscosities greatly exceed their molecular counterparts (see Eqs. [2.66] and [2.67]). In the astrophysical literature, it is usually accepted that one can write, for example, DV = LcVc, where Lc is some typical length and Vc is some typical speed of the turbulent motions. Unfortunately, although this expression is dimensionally correct, it is not possible at this writing to calculate unequivocally the quantities L c and Vc from results obtained on the basis of a linear stability analysis. A linear theory by its nature can say nothing about the process by which unstable eddylike or wavelike motions achieve some finite amplitude in the full nonlinear regime. Accordingly, no matter what kind of instability is assumed to be responsible for the small-scale motions, the magnitude of the eddy coefficients cannot quantitatively be given by a measure of the instability of the mean flow. That is to say, regardless of the spatial form that is assigned to the eddy coefficients, their overall magnitude can be determined only by fitting the chosen empirical formulae to the observational data.
It is not known at this writing whether one can find a better way of closing the equations for the large-scale flow in a rotating star. In any case, perhaps the greatest value of these parameterized models is that they give at least a reasonable global picture of the large-scale dynamics of the flow. They also provide a new perspective from which more elaborate models can be viewed.
3.7 Bibliographical notes
Section 3.2.1. The restriction imposed upon the angular velocity in a barotrope was originally derived by Poincare:
1. Poincare, H., Theorie des tourbillons, pp. 176-178, Paris: Georges Carre, 1893. An exhaustive discussion of barotropes and baroclines will be found in:
2. Wavre, R., Figuresplanetaires et geodesie, pp. 25-33, Paris: Gauthier-Villars,
Section 3.3.1. The reference to von Zeipel is to his paper:
3. von Zeipel, H.,Mon. Not. R. Astron. Soc., 84, 665, 1924.
The generalization to differentially rotating barotropes was made in:
An independent derivation of Vogt's result will be found in:
6. Roxburgh, I. W.,Mon. Not. R. Astron. Soc., 132, 201, 1966.
Section 3.3.2. The case of convective equilibrium was considered by:
7. Biermann, L., in Electromagnetic Phenomena in Cosmic Physics (Lehnert, B., ed.), I.A.U. Symposium No 6, p. 248, Cambridge: Cambridge University Press, 1958.
Kippenhahn's discussion is based on the Lebedinski-Wasiutynski equation, which is similar to our equation (3.45) but in which XV is replaced by 2fV — fi H). It was therefore concluded that anisotropic eddy viscosity should prevent uniform rotation. As was shown in Reference 34, however, the correct equation should depend on three independent parameters: fV, fH, and XV, which is not identically equal to 2(f V — fH). Since XV does not in general vanish in a convective zone, his method of solution thus remains essentially unchanged.
Sections 3.4.1 and 3.4.2. The classical references on the subject are those of:
9. Solberg, H., Proces-VerbauxAss. Meteor., U.G.G.I., 6eme Assemblee Generale (Edinburgh), Mem. et Disc., 2, 66, 1936.
10. H0iland, E., Archiv Mat. Naturv. (Oslo), 42, No. 5, 1, 1939.
11. H0iland, E., Avhandl. Norske Videnskaps-Akademi i Oslo,I,Mat.-Naturv. Klasse, No. 11, 1, 1941.
In their general form, conditions (3.94) and (3.95) were originally obtained in Reference 11. The analysis in these sections is taken from:
12. Fj0rtoft, R., Geofysiske Publikasjoner (Oslo), 16, No. 5, 1, 1946.
15. Lebovitz, N. R., Astrophys. J., 160, 701, 1970.
See especially Holmboe's paper. A preliminary discussion of finite-amplitude motions is given by:
16. Lorimer, G. S., and Monaghan, J. J., Proc. Astron. Soc. Australia, 4, 45, 1980.
Section 3.4.3. The barotropic and baroclinic instabilities are discussed at length in Reference 7 of Chapter 2. Interesting studies of nonaxisymmetric motions in baroclinic stars will be found in:
17. Fujimoto, M. Y., Astron. Astrophys., 176, 53, 1987.
18. Hanawa, T., Astron. Astrophys., 179, 383, 1987.
19. Fujimoto, M. Y., Astron. Astrophys., 198, 163, 1988.
Other pertinent comments on the literature will be found in Reference 26 (p. 392n) of Chapter 5.
Section 3.5. Reference is made to the following papers:
20. Goldreich, P., and Schubert, G., Astrophys. J., 150, 571, 1967.
22. Shibahashi, H., Publ. Astron. Soc. Japan, 68, 341, 1980.
Shibahashi's oscillatory instability is sometimes called "axisymmetric baroc-linic diffusive (ABCD) instability." (This is somewhat confusing, however, because the Goldreich-Schubert-Fricke instability is also an axisymmetric baroclinic diffusive instability.) The role of a /¿-gradient is further discussed in:
23. Knobloch, E., and Spruit, H. C., Astron. Astrophys., 125, 59, 1983.
Various evaluations of the time scale of the Goldreich-Schubert-Fricke instability will be found in:
24. Colgate, S. A., Astrophys. J. Letters, 153, L81, 1968.
25. Kippenhahn, R., Astron. Astrophys., 2, 309, 1969.
26. James, R. A., and Kahn, F. D., Astron. Astrophys., 5, 232, 1970; ibid., 12, 332, 1971.
27. Kippenhahn, R., Ruschenplatt, G., and Thomas, H. C., Astron. Astrophys., 91, 181, 1980.
28. Knobloch, E., Geophys. Astrophys. FluidDyn., 22, 133, 1982.
29. Korycansky, D. G., Astrophys. J., 381, 515, 1991.
See also Reference 16 (pp. 341-343) of Chapter 4. A rigorous derivation of Eq. (3.115) will be found in:
30. Lifshitz, A., and Lebovitz, N. R., Astrophys. J., 408, 603, 1993.
Section 3.6. Application of anisotropic eddy viscosity to astronomical problems was originally made by:
31. Lebedinski, A. I., Astron. Zh, 18, No. 1, 10, 1941.
32. Wasiutyriski, J., Astrophys. Norvegica, 4, 1, 1946.
The XV -effect has its roots in the work of Biermann:
33. Biermann, L., Zeit. Astrophys., 28, 304, 1951. A modern reference on these and related matters is:
34. Rudiger, G., Geophys. Astrophys. Fluid Dyn., 16, 239, 1980; ibid., 21, 1, 1982; ibid., 25,213, 1983.
See also Reference 7 of Chapter 5. The references to Schatzman and Press are to their papers:
35. Schatzman, E., Astron. Astrophys., 3, 331, 1969.
The following book is particularly worth noting:
37. Turner, J. S., Buoyancy Effects in Fluids, Cambridge: Cambridge University Press, 1973.
See also Reference 19. The inhibition of vertical mixing is further discussed in:
38. Vincent, A., Michaud, G., and Meneguzzi, M., Phys. Fluids, 8, 1312, 1996. For a lucid discussion of the eddy coefficients, see:
39. Canuto, V. M., and Battaglia, A., Astron. Astrophys., 193, 313, 1988.
Quite different empirical formulae for the eddy coefficients will be found in the current literature; see, for example:
40. Pinsonneault, M. H., Kawaler, S. D., and Demarque, P., Astrophys. J. Suppl, 74,501, 1990.
Compare Figure 16 in Reference 40 (p. 548) with the ad hoc formulae suggested in Reference 41. Such a comparison is useful because it clearly indicates that the practical evaluation of an eddy coefficient is at least partly an art, not just a science. Reference is also made to:
42. Balbus, S. A., and Hawley, J. F., Rev. Modern Phys., 70, 1, 1998.
Other papers dealing with magnetohydrodynamical effects in accretion disks and rotating stars may be traced to Reference 42. For the interested reader, a penetrating discussion of the weak-field shearing instability presented in Reference 42 (pp. 30-32) will be found in:
43. Acheson, D. J., and Hide, R., Rep. Prog. Phys., 36, 159, 1973. See especially their Section 4.3 (pp. 182-185).
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