## Meridional circulation

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...

## Applications to the Earths atmosphere

Since the atmosphere is essentially a thin layer of fluid on a sphere, a convenient set of axes at any point on the Earth's surface has x directed toward the east, y to the north, and z vertically upward (i.e., along the effective gravity ge, which combines the effects of the gravitational force and centrifugal force). If i, j, and k are unit vectors directed along these rotating axes, the relative velocity of the mean flow may be expressed as Letting ge - gk, one can rewrite the components of...

## Reynolds stresses and eddy viscosities

Laboratory experiments show that the transition from laminar to turbulent motions in an incompressible fluid depends on the Reynolds number which is a measure of the relative magnitude of the inertial to viscous forces occurring in the flow (see Eq. 2.7 ). Here U is the characteristic velocity of the flow, L is a characteristic length for the problem on hand, and v x p is the coefficient of kinematic viscosity. Turbulent flows always occur when the nondimensional number Re exceeds some critical...

## Circulation rotation and diffusion

It is generally thought that diffusion processes are responsible for most of the peculiar abundances observed in the chemically peculiar stars. As was originally noticed by Michaud (1970), abundance anomalies appear, on the main sequence, in the atmospheres of stars most likely to have stable envelopes and atmospheres. These stars are slow rotators and so have less meridional circulation, they often have magnetic fields, and they have an effective temperature for which stellar envelope models...

## The equations of fluid motion

Fluid dynamics proceeds on the hypothesis that the length scale of the flow is always taken to be large compared with the mean free path of the constitutive particles, so that the fluid may be treated as a continuum. This model makes it possible to treat fluid properties (such as velocity, pressure, density, etc.) at a point in space, with the physical variables being continuous functions of space and time. In other words, we assume that the macroscopic behavior of our systems is the same as if...

## Pm

Where j Q.m2 is the angular momentum per unit mass. Then, under what conditions is this configuration stable with respect to small isentropic disturbances Although no definitive answer can be given at the present time, some interesting results can be obtained for axially symmetric motions (i.e., motions for which the specific angular momentum of each fluid particle is preserved along its path). Departures from axial symmetry will be discussed briefly in Section 3.4.3. Two types of description...

## Subject index

A stars, axial rotation in, 11, 175, 176 ABCD instability, see Shibahashi oscillatory instability absolute magnitude, effect of rotation on, 169 acceleration of a fluid particle, 26, 29 age estimates of open clusters, effect of rotation on, 171 Am stars, axial rotation in, 12, 13, 173, 178-179 angular momentum diagram, 174-175 angular momentum, transport of, 93, 102, 151 anisotropic eddy diffusivity, see diffusivity anisotropic eddy viscosity, see viscosity Ap stars, axial rotation in, 12, 13,...

## Contact binaries The astrostrophic balance

In Sections 4.6 and 8.2-8.4 we have considered detached close binaries, in which the tidal distortions are relatively small and where components display physical characteristics that are similar to those of single stars. When the two components are separated by a few radii only, these tidal distortions may become, however, quite large. This is well illustrated by eclipsing binaries that exhibit sinusoidal-type light curves, and for which the first-order scheme of approximation adopted in the...

## FX

2 (n x u)H -R gradH T. (8.86) The necessity for large-scale astrostrophic currents also requires that we solve the nonlinear equation (8.69) for the temperature field in the common envelope. Here we have p0 Tu grad S div(x grad T). (8.87) By making use of Eqs. (2.11) and (8.83), we can also rewrite Eq. (8.87) in the more convenient form cVp0u grad T div(x grad T) , (8.88) where cV is the specific heat at constant volume. Equations (8.83), (8.86), and (8.88) are the fundamental equations of the...

## Close binaries

In Section 1.1 we pointed out that the early-type components of close binaries rotate more slowly than the average of single stars of the same spectral type. In contrast, whereas the rotational velocities of single main-sequence stars of spectral type F5 and later are quite small (i.e., less than 10 km s-1), appreciable rotations are common among the late-type components of close binaries. It has long been recognized that the distribution of rotational velocities in the close binaries is caused...

## Solar rotation

Until recently, only surface measurements of the solar rotation rate were available. Since the mid-1980s, with the advent of helioseismology, much has been learned about the internal rotation of the Sun through the inversion of -mode frequency splittings. As was noted in Section 1.2.2, it now appears that the observed surface pattern of differential rotation with latitude prevails throughout most of the solar convection zone, with equatorial regions moving faster than higher latitudes. In...

## D fi d p

Where f and j33 verify Eqs. (4.82)-(4.84) and the condition that both functions remain finite at r 0. From Eqs. (4.117) and (4.119) one readily sees that the large-scale motion consists of a constant overall rotation of O(e1 2), a meridional flow of O(e), and a back reaction of the currents of O(e3 2). The structure of this solution is very similar to that of a nondegenerate star. Of course, Eqs. (4.5) and (4.6) need to be modified. First, allowance must be made for a more general equation of...

## Axial rotation along the upper main sequence

In Section 1.3 we summarized the mean rotational properties of single stars. It is the purpose of this section to provide further information about the rotation patterns in specific groups of early-type, main-sequence stars. Figure 1.6 provides a comparison between the average rotational velocities of cluster and field stars. It is immediately apparent that the (v sin i) values of the, generally younger, cluster stars are similar to those of the field stars, except that for spectral types later...

## 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...