V x y

which also ensures that the net mass flux in the meridional direction exactly vanishes. Accordingly, this western boundary current returns northward a mass flux that precisely balances the southward Sverdrup mass flux. By virtue of the second equation (2.116), the northward velocity in this western boundary current is given by v = Vi(Xw, y) e-^sm^3 n- (2.132)

Both solutions were originally derived by Munk (1950). Figure 2.5 illustrates the zonal variation of the transport stream function, as given by V/ V, and the north-south velocity v across the western boundary layer. Note the intense northward flow and the small but significant counterflow just to the east of this boundary flow. This counterflow is actually found in observations of the Gulf Stream, and thus Munk's frictional model is qualitatively similar to the general oceanic circulation. (This large-scale flow possesses speeds of the order of 1-10 cm s-1, whereas the northward velocity in the Gulf Stream is typically 100 cm s-1, with a maximum speed of 200 cm s-1.) Unfortunately, one readily sees from Eq. (2.121) that the lateral scale of the flow is set by the horizontal eddy viscosity Kh, which is an adjustable parameter of the theory. Letting S ^ 50-100 km, which is the lateral dimension of the Gulf Stream, one finds that KV should be of the order of 107-108 cm2 s-1. Because such a value requires a very sizable frictional dissipation,

Fig. 2.5. The transport stream function, , and the north-south velocity v across the western boundary layer. The quantity v is measured in units of /S (see Eqs. [2.131] and [2.132]).

Fig. 2.5. The transport stream function, , and the north-south velocity v across the western boundary layer. The quantity v is measured in units of /S (see Eqs. [2.131] and [2.132]).

Bryan (1963) has made numerical calculations that retain both the frictional terms and the nonlinear terms (i.e., DZ /Dt) in Eq. (2.114). His nonlinear solutions, which require a smaller amount of lateral dissipation, converge to the purely frictional solution in the limit of large viscosities (e.g., Pedlosky [1987], pp. 309-311). As we shall see in Section 4.3, a similar problem arises in the discussion of large-scale meridional currents in stellar radiative zones.

2.7 Barotropic and baroclinic instabilities

Although a steady solution may prove useful for interpreting the general features of a large-scale motion, observations reveal that small-scale disturbances are inevitably present in any natural system. As a matter of fact, geophysicists have long recognized that the presence of fully developed disturbances - such as midlatitude cyclones - can be attributed to hydrodynamical instabilities. One form of instability that is of particular interest is called baroclinic instability. As was originally shown by Charney (1947), this mechanism depends on the presence of a vertical velocity shear in the mean zonal flow. By making use of Eq. (2.84), one readily sees that this shear requires the presence of horizontal temperature gradients. Since the surfaces of constant temperature and constant pressure do not coincide in such a system, it follows that the main source of energy for the baroclinic instability is the available potential energy that may be released and transferred to the small-scale disturbances. Another form of instability of geophysical interest depends on the presence of a horizontal velocity shear in the mean zonal flow. It is called barotropic instability, because it can occur in a system for which the surfaces of constant temperature and constant pressure do coincide. In the barotropic case, thus, the source of energy for the eddylike disturbances is associated with the kinetic energy stored in the mean flow. In more complex situations, however, these instabilities draw their energy from both the potential and kinetic energy of the basic state.

Deferring to Section 3.4 the study of these barotropic and baroclinic instabilities as they may occur in a rotating star, here we shall consider the simple geophysical model first introduced by Eady (1949). This configuration is particularly useful because it provides us with an overall perspective of the various kinds of instability that may arise in a rotating fluid.

The model neglects dissipative and curvature effects and uses the Boussinesq approximation for compressibility effects. Hence, we treat the density as a constant in all terms in the equations, except the one in the gravitational acceleration. Thus we assume a Boussinesq, isentropic, inviscid fluid, with constant density p and thermal expansion coefficient a. In the plane-parallel representation of Section 2.5, we interpret x as longitude, y as latitude, and z as height. The fluid is located on a plane rotating about the z axis with angular velocity f /2 and with a gravitational acceleration g. The system, which we assume to be in hydrostatic equilibrium in the z direction, is contained between the planes z = 0 and z = H and is unbounded horizontally. The basic state consists of the zonal wind u = U (z) and the potential temperature & = &(y, z), with 3&/3z > 0since we do not want to consider convective motions (see Eq. [2.16]). Hence, by making use of Eqs. (2.75) and (2.77), one can write fU =- - IT (2.133)

p dy and

p d z where f is regarded as a constant. Eliminating the deviation from hydrostatic pressure by cross-differentiation, we obtain the thermal wind relation dU d&

dz dy which relates the vertical velocity shear to the latitudinal potential-temperature gradient (see Eq. [2.84]).

At this juncture, it is convenient to define the squared buoyancy frequency

d z which is a measure of the stability of the fluid layer against vertical disturbances. (When N2 > 0, the frequency N merely corresponds to buoyancy oscillations, i.e., stable gravity modes; when N2 < 0, it corresponds to rising or sinking motions.) N is usually called the Brunt-Vaisala frequency. We shall also define the nondimensional number

which is known as the Richardson number. A positive value of this number corresponds to a convectively stable stratification in the vertical direction.

Equations (2.133)-(2.135) define the unperturbed state of the flow. We now assume small deviations from this basic state. Hence, we linearize Eqs. (2.73)-(2.75), (2.77), and (2.12). Thus neglecting nonhydrostatic effects in the z component of the momentum equation, one obtains the following set of equations:

du 1 dv1 9wi

d t d x J dz p d x d d \ 1 dp1 - + U — V1 + fui =--p1, (2.140)

1 dp1

where ui, vi, Ti, pi, and ©i designate, respectively, the small Eulerian changes in the velocity components, pressure, and potential temperature.

We now consider a zonal flow of magnitude u0 and constant vertical shear u0/H and a potential temperature with constant stratification d©/dz and constant horizontal gradient d©/dy. For convenience, we shall make use of the following basic units: x ^ u0/f, y ^ u0/f, z ^ H, t ^ 1/f, U ^ u0, © ^ H(9©/9z), u1 ^ u0, v1 ^ u0, and w1 ^ fH. In these units, our basic steady state becomes

where Ri = ag(d©/dz)/(u0/H)2 is a constant. Since the coefficients of the linearized equations depend on z only, we can thus let w1 = W (z)exp[i (ct t + kx + ly)], (2.145)

and we can write similar expressions for the other Eulerian variations in Eqs. (2.138)-(2.142). Given our choice of units, we have k ^ f/u0, l ^ f/u0, and ct ^ f. With a little algebra, these equations can be reduced to the following equation for the function W:

We shall also prescribe that

Equations (2.146) and (2.147) constitute an eigenvalue problem for the complex frequency o = or + i. Since the unit of horizontal scale is u0/f, the nondimensional wavenumbers k and l are simply the zonal and latitudinal Rossby numbers of the perturbations (see Eq. [2.30]).

A detailed study of this eigenvalue problem has been made by Stone (1966, 1970, 1972), who integrated Eqs. (2.146) and (2.147) for a wide range of values for the nondimensional parameters k, l, and Ri. His analysis shows that three basically different instabilities can occur for strictly positive Richardson numbers: a symmetric instability of the kind discussed by Solberg and Hailand (see Section 3.4.2), a baroclinic instability of the kind first discussed by Charney (1947) and Eady (1949), and a shear-flow instability analogous to the Kelvin-Helmholtz instability of two superposed fluids with different velocities and densities.* Not unexpectedly, because there is no latitudinal shear in the mean zonal flow U = z, barotropic instability does not occur in this simple model (see, however, Section 3.4.3).

2.7.1 The symmetric instability

The maximum growth rates for this instability are associated with perturbations that correspond to k = 0, l ^ 1, and large growth rates (\ai | ^ 1). Letting k = 0 in Eq. (2.146), one obtains

, d2W dW (1 - a2)—~r~r + 2il —--12RiW = 0. (2.148)

dz2 dz

Its solution, subject to boundary conditions (2.147), is

2 \mn with m = 1, 2, 3, Only the eigenvalues corresponding to the minus sign in front of the square root may lead to unstable motions. By virtue of Eq. (2.150), this instability occurs if and only if

The growth rate |oi | is maximum for l = <x> and has the value

* See, e.g.,Chandrasekhar, S.,Hydrodynamic andHydromagneticStability, Sections 100-104, Oxford: Clarendon Press, 1961 (New York: Dover Publications, 1981); Drazin, P. G., and Reid, W. H., Hydrodynamic Stability, Section 44, Cambridge: Cambridge University Press, 1981.

By making use of Eqs. (2.135) and (2.137), one readily sees that this instability may be visualized as the response to a large horizontal potential-temperature gradient in the form of an axisymmetric motion (k = 0) with small latitudinal wavelengths (l ^ 1), that is, a series of rolls parallel to the mean zonal flow. As was shown by Stone (1972), these motions draw their energy from both the kinetic and potential energy of the basic flow. For the most unstable modes (l ^ro), however, the potential energy release is negligible. Accordingly, this instability may also be viewed as a form of barotropic instability. The link between condition (2.151) and the Solberg-H0iland conditions will be established at the end of Section 3.4.2.

2.7.2 The baroclinic instability

The maximum growth rates for this instability are associated with perturbations that are independent of latitude (l = 0), have large zonal scales (k ^ 1), and have small growth rates (|Oi | ^ k). Thus we let l = 0 and o = kc in Eq. (2.146) to obtain

d2 W

2 dW

Since the largest growth rates are found in the range 0 < k ^ 1, we shall expand the solutions of the eigenvalue problem in powers of k2. Letting

W = W0 + k2 W1 + in Eq. (2.153), one can show that and c = c0 + k2c1 +

Wi = 3cj(co + z)2 + — c3(co + z)2 + —^ (co + z)5, etc. Applying boundary conditions (2.147) to these solutions, we obtain*

Hence, the growth rates of this instability can be written approximately as

Ignoring terms of order k5 or higher, one finds that the most rapidly growing perturbation is the one with the wavenumber iki=i ^r

* In this simple mathematical model, which does not include the latitudinal variation of the Coriolis parameter (i.e., the fi term in Eq. [2.78]), one thus finds a cutoff wavelength, below which all disturbances are stable, and above which those of larger scale are unstable. As was originally shown by Green (1960), however, when Eady's (1949) problem is modified by taking f > 0, the flow becomes unstable to disturbances of all wavelengths, even for small values of f. In a more realistic formulation of baroclinic instability, there is thus no short wave limit for the instability region of the wave spectrum (e.g., Pedlosky [1987], Fig. 7.8.4).

and growth rate

One can show that this instability dominates over the symmetric instability whenever Ri > 1. Stone's (1972) analysis also shows that the kinetic energy of the growing perturbations is drawn from both the potential and kinetic energy of the basic state. When Ri ^ 1, however, the kinetic energy release is negligible compared to the potential energy release. This is the reason why this instability is called a baroclinic instability. It will be discussed further in Section 3.4.3.

2.7.3 The shear-flow instability

This instability is associated with relatively small-scale perturbations (k ^ 1) and has small growth rates ( | ai | ^ k). Again, letting a = kc in Eq. (2.146) and assuming that at most l ^ k in the limit k ^<x>, one finds that Eq. (2.146) reduces to

Following Stone (1966), we obtain the solution for this equation:

W = (c + z)1/2[ ^(c + z )q + B (c + z)-q ], (2.162)


Applying boundary conditions (2.147), one finds that anontrivial solution exists if c = 0, c = 1, or

2 2 2q with m = 1, 2, 3,____ This equation shows that unstable solutions exist if q is real. In particular, q will be real inside the region l2 < k2^^ - , k » 1, (2.165)

and such a region exists as long as

Equation (2.165) shows that this instability is greatest for small latitudinal wavenumbers, with the perturbations consisting of a series of rolls perpendicular to the mean zonal flow. Like the symmetric instability, it is also a form of barotropic instability, because it draws its energy mainly from the kinetic energy of the basic flow, although it may also store up potential energy. It will be considered further in Section 3.4.3.

2.8 Self-gravitating fluid masses

In this section we shall consider some general properties of self-gravitating bodies rotating freely in space. As we shall see in Section 6.2.1, although the interest of these models may be of a rather formal character, some of them have a direct bearing on the internal structure of rotating stars.

2.8.1 The virial equations

If we neglect friction altogether, Eq. (2.7) can be rewritten in the form

Dt dxk p dxk

Jv |r - r'j where G is the constant of gravitation, V is the total volume of the configuration, and dv is the volume element.

Multiply now the left-hand side of Eq. (2.167) by pxt and integrate over the entire volume. By virtue of mass conservation, one has

Dvk d

lv Dt dt jv where f Dvk d f

Dt dt

(see Eq. [3.53] below). Similarly, by making use of Eq. (2.168), we can write i pXi dVdv = G ( f p(r, t) p(r', t) ^(Xk -X) dv dv'. (2.171)

Thus, if we let

Wk = -1 G [ ( p(r, t) p(r', t)(Xi ^xi)(xk 3 xk) dv dv', (2.172) 2 JvJv |r - r'|3

Finally, the last term in Eq. (2.167) can be integrated by parts to give

Iv dxk Jv since the pressure must vanish on the free surface.

Combining Eqs. (2.169), (2.173), and (2.174), we obtain d dt j v

Jv Jv

Since all tensors on the right-hand side are symmetric, it follows that the left-hand side must also be symmetric. Hence, we can write d r d r

dt Jv dt Jv

This equation, which embodies the conservation of the total angular momentum J, implies that d r 1 d2 r

dt Jv 2 dt2 Jv

Equation (2.175) thus becomes

1 d2 Iik

2 dt2 Jv where

These are the second-order virial equations in their usual form.

By contracting the indices in Eq. (2.178), we obtain the scalar virial equation,

2 dt2 Jv where I is the moment of inertia with respect to the center of mass and K is the total kinetic energy. By virtue of Eqs. (2.168) and (2.172), one also has

2 J v which is the gravitational potential energy.

For a self-gravitating fluid that rotates steadily about the x3 axis with some assigned angular velocity Q, Eq. (2.180) becomes

K = 1 f pQ2 (xj + X22) dv, (2.183) 2 J v which is the rotational kinetic energy. Since the volume integral over the pressure always remains a nonnegative quantity, it follows at once that the ratio t = K/|W | (2.184)

is limited by equilibrium requirements to range from t = 0 (a spherical body) to t = 0.5. Of course, at this stage we do not know a priori whether the entire domain of values for t (or J) can be covered by suitable models.

2.8.2 The Maclaurin and Jacobi ellipsoids

From a purely mechanical point of view, the specification of a particular model in a state of permanent rotation depends on three quantities: (i) the total mass M, (ii) the total angular momentum J, and (iii) an assigned distribution for the angular momentum per unit mass Q(xj2 + x|). To clarify some aspects of the general problem, let us consider uniformly rotating, homogeneous ellipsoids. For that purpose, consider a system that is at rest with respect to a frame of reference rotating with the constant angular velocity Q. The equations of relative equilibrium referred to rectangular axes rotating around the x3 axis are

(i = 1, 2, 3; no summation over repeated indices). Now, the components of the gravitational attraction in a homogeneous ellipsoid (with semi-axes a1, a2, and a3) have the form d V

d Xi where rdu

By virtue of Eq. (2.186), the three components of Eq. (2.185) can be readily integrated to give p 1 ( ) ( )

— = - Q2 (x2 + x2) - nGp (Aix2 + A2x2 + A3xj) + constant, (2.189) p2

so that the surfaces of constant pressure take the form

A1--— x,2 + A2--— x2 + A3 x2 = constant. (2.190)

In expressing that the boundary of the ellipsoid,

222 xxx

a12 a22 a32

coincides with one of the surfaces defined in Eq. (2.190), one finds that

Hence, we must have a2a22(A1 - A2) + (a2 - a22)a32A3 = 0 (2.193)

Q 2 a12 A 1 - a22 A 2 a12 A 1 - a32 A 3 a22 A 2 - a32 A 3

Obviously, the first equality (2.194) obtains only if a\ = a2 = a3. If we next make use of Eq. (2.187), Eq. (2.193) becomes

2 2 a2a2

(a2 + u) (a| + u) a| + u Finally, the three equalities (2.194) lead to the following relations:

when a1 = a2 = a3; without any restriction, we also find

and a similar expression in which the index 1 replaces the index 2, and conversely.

From Eq. (2.197) and its unwritten companion, we first observe that a\ > a3 and a2 > a3. Thus, the rotation must always take place about the least axis. However, we may have either a\ > o2 or a\ < o2, since there is no physical difference between any two configurations for which we exchange the indices 1 and 2. Finally, we perceive at once that Eq. (2.195) can be satisfied in two different ways. Either we let a\ = a2 or, whenever possible, we let a\ > a2 (say) and make the integral factor vanish in Eq. (2.195). The former solution defines the Maclaurin spheroids while the latter corresponds to the Jacobi ellipsoids* The Maclaurin spheroids range from a sphere (t = 0) to an infinitely thin disk that is at rest (t = 0.5). A numerical integration of Eq. (2.195) reveals that the Jacobi ellipsoids exist only in the domain Tb < t < 0.5, where Tb = 0.1375; they range from the bifurcation spheroid (t = Tb, where a3/a\ = 0.5827) to an infinitely long needle that is devoid of rotational motion (t = 0.5). Figure 2.6 illustrates the behavior of Q.2 as a function of t. Thus, when 0 < t < Tb, the Maclaurin spheroids are the only possible figures of equilibrium; in contrast, in the range Tb < t < 0.5, to each value of t correspond two ellipsoidal configurations in relative equilibrium: one Maclaurin spheroid and one Jacobi ellipsoid.

For fixed values of J, M, and V, one can show that the total mechanical energy K + W is smaller in the body with triplanar symmetry than in the corresponding axisymmetric oo a

* The Scottish mathematician Colin Maclaurin (1698-1746) was the first to show in 1740 that any oblate homogeneous spheroid is a possible figure of equilibrium for uniformly rotating bodies. The next important discovery was not made until 1834, however, when the German mathematician Carl Jacobi (1804-1851) pointed out that homogeneous ellipsoids with three unequal axes can very well be figures of equilibrium (Poggendorff Ann., 33, 229, Oct. 4, 1834). Competition was fierce then, as it is today. Indeed, about three weeks later Joseph Liouville (1809-1882) published the detailed analytical proof of that theorem; however, noting that Jacobi had merely reported Eqs. (2.195) and (2.196) in his paper, Liouville could not refrain from saying that "this theorem, simple as it is, seems to have been enunciated as a challenge to the French mathematicians." And then Liouville added: "Mr. Jacobi was promising more indeed, when he announced that he was going to take over celestial mechanics from the pitiful state in which, so he said, Laplace had left it" (J. Ecole Polytech. Paris, 14, Cahier23, p. 291n, Oct. 27, 1834). Thus, Liouville perceived at once -but was reluctant to admit - that Jacobi had made an important discovery; yet, none of them could have foreseen that they were discussing the first known case of broken symmetry in physics. For the interested reader, the above quotations from Liouville's paper should clarify Chandrasekhar's (1969, p. 7) presentation of the Jacobi ellipsoids.

2.8 Self-gravitating fluid masses .25 | i i i i | i i i i | i i

Cvl a

2.8 Self-gravitating fluid masses .25 | i i i i | i i i i | i i

Fig. 2.6. The squared angular velocity Q2 along the Maclaurin (solid line) and the Jacobi (dashed line) sequences, as a function of the ratio t = K /\W |. The unit of Q2 is 2n Gp.

configuration. Accordingly, if some dissipative mechanism is operative, we may expect that beyond the point of bifurcation, t = tb, an incompressible Maclaurin spheroid will evolve gradually to the Jacobi ellipsoid having the same total angular momentum. A detailed study of the global oscillations that transform a Maclaurin spheroid into agenuine triaxial body while preserving its plane of symmetry is thus in order.

Such a study was made by Lebovitz (1961). In particular, by making use of the perturbation forms of the six components of Eq. (2.178), he was able to calculate the five oscillation frequencies that reduce to the quintuple Kelvin frequency belonging to the spherical harmonics Y2m in the limit Q = 0 (\m \ < 2). As we shall see, it is the toroidal (or barlike) modes that are of direct relevance to our discussion. When dissipative effects may be neglected, the corresponding frequencies are

This is an exact analytical result. Figure 2.7 illustrates the behavior of the frequencies a-2 and o+2 along the Maclaurin sequence. We observe that o-2 vanishes when Q2 = m, that is, at the point t = tb where the Jacobi sequence branches off the Maclaurin sequence. In addition, both frequencies become complex when Q2 > 2m, that is, beyond the point t = t = 0.2738, where a3/a1 = 0.3033; clearly, this implies instability by an overstable oscillation of frequency Q.

ct-2 = Q - (2m - Q2)1/2 and a+2 = Q + (2m - Q2)1/2, and two similar frequencies in which — Q replaces Q. Here we have let

ct-2 = Q - (2m - Q2)1/2 and a+2 = Q + (2m - Q2)1/2, and two similar frequencies in which — Q replaces Q. Here we have let

Fig. 2.7. Frequencies of the barlike modes (\a-2\ and \o+2\) along the Maclaurin sequence, as functions of the ratio t = K/\W\. The frequencies are given in units of (4n Gp)1/2; they are not represented beyond the point t = Tt where they become complex. After Lebovitz (1961). Source: Ostriker, J. P., and Bodenheimer, P., Astrophys. J., 180, 171, 1973.

The situation is somewhat different when dissipation is properly taken into account. In that case, one can show that the barlike modes of oscillation are damped prior to the neutral point t = Tb .In the range Tb <t <xt, however, the slightest amount of dissipation will carry slowly the Maclaurin spheroid into another configuration having a genuine triplanar symmetry. The system is then said to be secularly unstable. Beyond the point t = t , the Maclaurin spheroid becomes dynamically unstable as in the nondissipative case.

2.8.3 Rotating polytropes

In Section 2.8.2 we have summarized the main properties of the Maclaurin-Jacobi ellipsoids. To what extent can we extrapolate these results to centrally condensed bodies that we force to rotate with some prescribed angular momentum distribution? In particular, is it always possible to build a model in a state of permanent rotation for all values that we may assign to the total angular momentum J? To illustrate these problems, let us briefly consider polytropes, that is, barotropic structures for which the pressure p and the density p are related by the relation p = K)P1+1/", (2.200)

where K0 and n are constants (0 < n < 5). Because the value n = 0 corresponds to a configuration of uniform density, it is therefore convenient to think of the incompressible Maclaurin spheroids as a sequence of uniformly rotating polytropes of index n = 0.

It has been known since the pioneering work of Jeans (1919), which was subsequently refined by James (1964), that every sequence of axially symmetric, uniformly rotating

Fig. 2.7. Frequencies of the barlike modes (\a-2\ and \o+2\) along the Maclaurin sequence, as functions of the ratio t = K/\W\. The frequencies are given in units of (4n Gp)1/2; they are not represented beyond the point t = Tt where they become complex. After Lebovitz (1961). Source: Ostriker, J. P., and Bodenheimer, P., Astrophys. J., 180, 171, 1973.

polytropes terminates with a configuration in which the gravitational and centrifugal accelerations exactly balance at points on the equator. As a rule, each sequence terminates at a point t = Tmax (say), and the values of Tmax decrease sharply with polytropic index. (While Tmax = 0.5 along the Maclaurin sequence, Tmax ^ 0.12 when n = 1, and Tmax is already reduced to less than one percent when n = 3.) To be specific, for low polytropic index (i.e., n < 0.8) sequences of axisymmetric models reach points of bifurcation and, where further models become secularly unstable (as for n = 0), the sequences may bifurcate into analogs of the Jacobi ellipsoids. When n > 0.8, however, sequences of axially symmetric, uniformly rotating polytropes always terminate at models in which the effective gravity vanishes at the equator.

In the early 1980s, different groups have actually constructed complete sequences of nonaxisymmetric, uniformly rotating polytropes for n < 0.8. Their independent calculations clearly show that all these Jacobi-like sequences bifurcate from their corresponding axisymmetric counterparts at about the same value of t (^ 0.137). However, as was shown by Eriguchi and Hachisu (1982), even for an index as low as n = 0.1, they terminate after only a small increase in J away from the axisymmetric models. Accordingly, until limited by the onset of equatorial breakup, the equilibrium figures of uniformly rotating polytropes with low polytropic index resemble, in all essential respects, the Maclaurin-Jacobi ellipsoids. This is in sharp contrast to the more centrally condensed polytropes for which the rotational kinetic energy does not exceed a small fraction of the gravitational potential energy (t ^ 0.5). Indeed, as was shown by Tassoul and Ostriker (1970), because uniformly rotating configurations having a genuine triplanar symmetry always branch off at the point t ^ 0.137, which is almost independent of n, bifurcation - and the ensuing secular instability - does not occur along polytropic sequences with n > 0.8. In other words, uniformly rotating, centrally condensed polytropes remain unaffected by the global instabilities described in Section 2.8.2 because they cannot store a large amount of angular momentum!

As was originally shown by Bodenheimer and Ostriker in 1973, a completely different picture emerges from the study of frictionless, differentially rotating polytropes for which we prescribe a given angular momentum distribution. In that case, the centrally condensed models closely simulate incompressible Maclaurin spheroids, except that they do not maintain uniform rotation. In particular, it was found that the polytropic sequences do not terminate and that bifurcation may occur when T is very closely equal to the value t = Tb obtained for the Maclaurin spheroids. Moreover, their work strongly suggested that dynamical instability with respect to a barlike mode always sets in beyond the point t = Ti, which again does not greatly depend on the particular sequence. Recent developments have shown that some of these propositions may need refinement, however.

To be specific, Imamura et al. (1995) have shown that for angular momentum distributions similar to those of the Maclaurin spheroids, there is a qualitative correspondence between the onset of secular instability for compressible and incompressible fluids. However, for angular momentum distributions that are more peaked toward the equatorial radius, their work indicates that secular instability with respect to a barlike mode sets in at lower values of t, shifting from t = 0.14 to t = 0.09 over the range of angular momentum distributions considered. More recently, Toman et al. (1998) have shown that the onset of dynamical instability with respect to a barlike mode is not very sensitive to the compressibility or angular momentum distribution when the polytropic models are parameterized by t . The eigenfunctions for the fastest growing barlike modes are, however, qualitatively different from the Maclaurin eigenfunctions in one important respect: They develop strong spiral arms. These spiral arms are stronger for larger values of the polytropic index and for configurations whose angular momentum distributions deviate significantly from those of the Maclaurin spheroids.

2.9 Bibliographical notes

The literature on classical hydrodynamics is very extensive. Among the many textbooks on the subject, my own preference goes to:

1. Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge: Cambridge University Press, 1967.

2. Landau, L. D., and Lifshitz, E. M., Fluid Mechanics, 2nd Edition, Oxford: Perg-amon Press, 1987.

3. Tritton, D. J., Physical Fluid Dynamics, 2nd Edition, Oxford: Clarendon Press, 1988.

The only book devoted exclusively to the problem of rotation is:

4. Greenspan, H. P., The Theory of Rotating Fluids, Cambridge: Cambridge University Press, 1968 (reprinted by Breukelen Press, Brookline, MA, 1990).

Excellent introductions to geophysical fluid dynamics are:

5. Houghton, J. T., The Physics of Atmospheres, 2nd Edition, Cambridge: Cambridge University Press, 1986.

6. Holton, J. R., An Introduction to Dynamic Meteorology, 3rd Edition, New York: Academic Press, 1992.

At a more advanced level, the following monograph is particularly worth noting:

7. Pedlosky, J., Geophysical Fluid Dynamics, 2nd Edition, New York: SpringerVerlag, 1987.

See also:

8. Gill, A. E., Atmosphere-Ocean Dynamics, Orlando: Academic Press, 1982.

9. Apel, J. R., Principles of Ocean Physics, Orlando: Academic Press, 1987.

A general survey from the viewpoint of astrophysics will be found in:

10. Shore, S.N., An Introduction to Astrophysical Hydrodynamics, San Diego: Academic Press, 1992.

Sections 2.5 and 2.6. Ekman layers, at a rigid plane boundary and at the ocean-atmosphere interface, were originally discussed in:

11. Ekman, V. W., ArkivMat. Astron. Fysik (Stockholm), 2, No 11, 1, 1905.

However, the first quantitative discussions of the so-called Ekman pumping mechanism were given by:

12. Bondi, H., and Lyttleton, R. A., Proc. Cambridge Phil. Soc., 44, 345, 1948.

13. Charney, J. G., and Eliassen, A., Tellus, 1, No 2, 38, 1949.

A comprehensive discussion of Ekman pumping is given by:

14. Greenspan, H. P., and Howard, L. N., J. FluidMech., 17, 385, 1963. An illustrative example will be found in:

15. Harada, A., J. Meteorol. Soc. Japan, 60, 876, 1982. Oceanic currents are discussed in:

16. Sverdrup, H. U., Proc. Natl. Acad. Sci. U.S.A., 33, 318, 1947.

See also Reference 7 for a comprehensive review of these matters.

Section 2.7. Reference is made to the following papers:

21. Green, J. S. A., Quart. J. Roy. Meteorol. Soc., 86, 237, 1960.

The analysis in this section is largely derived from:

22. Stone, P. H., J. Atmos. Sci., 23, 390, 1966; ibid, 27, 721, 1970; ibid, 29, 419, 1972.

Deviations from hydrostatic equilibrium in Eq. (2.141) are discussed in:

23. Tokioka, T., J. Meteorol. Soc. Japan, 48, 503, 1970.

The mass-divergence effect in the continuity equation was also considered by:

25. Hyun, J. M.,andPeskin,R. L., J. Atmos. Sci.,33,2054,1976;ibid,35,169,1978.

Sections 2.8.1 and 2.8.2. The standard reference on the virial equations and homogeneous ellipsoids is:

26. Chandrasekhar, S., Ellipsoidal Figures of Equilibrium, New Haven: Yale University Press, 1969 (reprinted, with Section 63 revised, by Dover Publications, New York, 1987).

The reference to Lebovitz is to his paper:

27. Lebovitz, N. R., Astrophys. J., 134, 500, 1961.

With the advent of fast computers, several authors have computed new equilibrium sequences that branch off the Maclaurin-Jacobi ellipsoids and have finite deformations from these configurations. The following papers are particularly worth noting:

28. Eriguchi, Y., and Hachisu, I., Prog. Theor. Phys, 67, 844, 1982.

29. Eriguchi, Y., Hachisu, I., and Sugimoto, D., Prog. Theor. Phys., 67,1068,1982.

30. Eriguchi, Y., and Hachisu, I., Astron. Astrophys., 148, 289, 1985.

See also Reference 33 (pp. 544-546).

Section 2.8.3. For a general account of rotating polytropes, see:

31. Tassoul, J. L., Theory of Rotating Stars, pp. 233-272, Princeton: Princeton University Press, 1978.

Other useful reviews of rotating fluid masses can be found in:

32. Lebovitz, N. R., Annu. Rev. FluidMech., 11, 229, 1979.

33. Durisen, R. H., and Tohline, J. E. in Protostars & Planets II (Black, D. C., and Matthews, M. S., eds.), p. 534, Tucson: University of Arizona Press, 1985.

The following references are quoted in the text:

34. Jeans, J. H., Problems of Cosmogony and Stellar Dynamics, pp. 165-186, Cambridge: Cambridge University Press, 1919.

36. Tassoul, J. L., and Ostriker, J. P., Astron. Astrophys., 4, 423, 1970.

37. Bodenheimer, P., and Ostriker, J. P., Astrophys. J., 180, 159, 1973.

38. Ostriker, J. P., and Bodenheimer, P., Astrophys. J., 180, 171, 1973.

Sequences of uniformly rotating, nonaxisymmetric polytropes with n < 0.8 have been constructed in:

39. Ipser, J. R., and Managan, R. A., Astrophys. J., 250, 362, 1981.

40. Vandervoort, P. O., and Welty, D. E., Astrophys. J., 248, 504, 1981.

41. Hachisu, I., and Eriguchi, Y., Prog. Theor. Phys., 68, 206, 1982.

There is also a growing literature on the nonaxisymmetric instabilities in differentially rotating polytropes. The following papers may be noted:

42. Imamura, J. N., Toman, J., Durisen, R. H., Pickett, B. K., and Yang, S., Astrophys. J., 444,363, 1995.

43. Pickett, B. K., Durisen, R. H., and Davis, G. A., Astrophys. J., 458, 714, 1996.

44. Smith, S. C., Houser, J. L., and Centrella, J. M., Astrophys. J., 458, 236, 1996.

45. Toman, J., Imamura, J. N., Pickett, B. K., and Durisen, R. H., Astrophys. J., 497, 370, 1998.

Other contributions may be traced to References 42-45.

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