The

In Section 1.1 we briefly discussed the early measurements of the axial rotation of the Sun. With the advent of more sensitive instruments, however, Doppler and tracer measurements have shown that the solar atmosphere exhibits motions on widely different scales. Besides the large-scale axisymmetric motions corresponding to differential rotation and meridional circulation, velocity fields associated with turbulent convection and also with oscillatory motions at about a five-minute period have been observed. Considerable attention has focused on analysis of these oscillations since, for the very first time, they make it possible to probe the Sun's internal rotation.

1.2.1 Large-scale motions in the atmosphere

The solar surface rotation rate may be obtained from measurements of the longitudinal motions of semipermanent features across the solar disk (such as sunspots, faculae, magnetic field patterns, dark filaments, or even coronal activity centers), or from spectrographic observations of Doppler displacements of selected spectral lines near the solar limb. Each of the two methods for deriving surface rotation rates has its own limitations, although few of these limitations are common to both. Actually, the determination of solar rotation from tracers requires that these semipermanent features be both randomly distributed throughout the fluid and undergo no appreciable proper motion with respect to the medium in which they are embedded. In practice, no tracers have been shown to possess both characteristics; moreover, most of them tend to occur in a limited range of heliocentric latitudes. By the spectrographic method, rotation rates can be found over a wider range of latitudes. But then, the accuracy is limited by the presence of inhomogeneities of the photospheric velocity field and by macroscopic motions within coronal and chromospheric features, so that the scatter between repeated measurements is large.

Figure 1.1 assembles sidereal rotation rates obtained from photospheric Doppler and tracer measurements. The observations refer to the sunspots and sunspot groups, magnetic field patterns, and Doppler shifts. In all cases the relationships shown in Figure 1.1 are

Fig. 1.1. Comparison of the solar differential rotation obtained by different methods. Source: Howard, R., Annu. Rev. Astron. Astrophys., 22, 131, 1984. (By permission. Copyright 1984 by Annual Reviews.)

smoothed curves obtained by fitting the data to expansions in the form

The decrease of angular velocity with increasing heliocentric latitude is clear. However, it is also apparent that different techniques for measuring the solar surface rotation rate yield significantly different results. In particular, the sunspot groups rotate more slowly in their latitudes than individual sunspots. Note also that the rotation rate for the magnetic tracers is intermediate between that for the individual spots and that for the photospheric plasma. It is not yet clear whether these different rotation rates represent real differences of rotation at various depths in the solar atmosphere or whether they reflect a characteristic behavior of the tracers themselves.

Chromospheric and coronal rotation measurements have also been reported in the literature. It seems clear from these results that the latitudinal gradient of angular velocity depends very much on the size and lifetime of the tracers located above the photosphere. To be specific, the long-lived structures exhibit smaller gradients than the short-lived ones, and the very long-lived coronal holes rotate almost uniformly. These noticeable differences remain poorly understood.

Fig. 1.2. Residuals of annual average sunspot rotation rates for the period 1921-1982. Solar cycle maxima timing and length are denoted by numbered boxes. Vertical lines denote year of sunspot minimum. Source: Gilman, P. A., and Howard, R., Astrophys. J, 283, 385, 1984.

Figure 1.1 merely illustrates the mean properties of the solar surface differential rotation. As was originally shown by Howard and LaBonte (1980), however, analysis of the residual motions in the daily Doppler measurements made at Mount Wilson suggests the presence of a torsional oscillation of very small amplitude in the photosphere. This oscillation is an apparently organized pattern of zonally averaged variations from a mean curve for the differential rotation, as defined in Eq. (1.3). The amplitude of the residuals constituting the torsional oscillation is of the order of 5 m s-1. It is a traveling wave, with latitude zones of fast and slow rotation, that originates near the poles and moves equatorward over the course of a 22-year cycle. The latitude drift speed of the shear is of the order of 2 m s-1. In the lower heliocentric latitudes, the torsional shear zone between the fast stream on the equator side and the slow stream on the pole side is the locus of solar activity. This coincidence strongly suggests that this torsional oscillation is somewhat related to the solar activity cycle.

Variations of the solar surface rotation rate over individual sunspot cycles have been reported by many investigators. Detailed analyses of the Mount Wilson sunspot data for the period 1921-1982 suggest that on average the Sun rotates more rapidly at sunspot minimum. * A similar frequency of rotation maxima is also seen in the Greenwich sunspot data for the years 1874-1976. The variability of the mean rotation rate is illustrated in Figure 1.2, which exhibits peaks of about 0.1 degree day-1 in the residuals near minima of solar activity. The Mount Wilson data also show variations from cycle to cycle, with the most rapid rotation found during cycles with fewer sunspots and less sunspot area.

* A similar result was obtained by Eddy, Gilman, and Trotter (1977) from their careful analysis of drawings of the Sun made by Christopher Scheiner (during 1625-1626) and Johannes Hevelius (during 1642-1644). During the earlier period, which occurred 20 years before the start of the Maunder sunspot minimum (1645-1715), solar rotation was very much like that of today. By contrast, in the later period, the equatorial velocity of the Sun was faster by 3 to 5% and the differential rotation was enhanced by a factor of 3. These results strongly suggest that the change in rotation of the solar surface between 1625 and 1645 was associated, as cause or effect, with the Maunder minimum anomaly.

Very recently, Yoshimura and Kambry (1993) have found evidence for a long-term periodic modulation of the solar differential rotation, with a time scale of the order 100 years. This modulation was observed in the sunspot data obtained by combining Greenwich data covering the period 1874-1976 and Mitaka data covering the period 1943-1992. Their analysis suggests that there exists a well-defined periodic variation in the overall rotation rate of the photospheric layers. To be specific, it is found that the surface rotation rate reaches a maximum at solar cycle 14, decreases to a minimum at cycle 17, and increases again to reach a maximum at cycle 21. Moreover, the time profile of the long-term modulation of the solar rotation is quite similar to the time profile of the solar-cycle amplitude modulation, but the two profiles are displaced by about 23 years in time. Further study is needed to ascertain whether this long-term modulation is strictly periodic or part of a long-term aperiodic undulation.

Several observational efforts have been made to detect a mean north-south motion on the Sun's surface. Unfortunately, whereas the latitudinal and temporal variations of the solar rotation are reasonably well established, the general features of the meridional flow are still poorly understood. Three different techniques have been used to measure these very slow motions: (i) the Doppler shift of selected spectral lines, (ii) the displacement of magnetic features on the solar disk, and (iii) the tracing of sunspots or plages. A majority of Doppler observations suggests a poleward motion of the order of 10 m s-1, whereas others differ in magnitude and even in direction. Doppler data obtained with the Global Oscillation Network Group (GONG) instruments in Tucson from 1992 to 1995 indicate a poleward motion of the order of 20 m s-1, but the results also suggest that the Sun may undergo episodes in which the meridional speeds increase dramatically. The analysis of magnetic features shows the existence of a meridional flow that is poleward in each hemisphere and is of the order of 10 m s-1, which agrees with most of the Doppler measurements. On the contrary, sunspots or plages do not show a simple poleward meridional flow but a motion either toward or away from the mean latitude of solar activity, with a speed of a few meters per second. Analysis of sunspot positions generally shows equatorward motions at low heliocentric latitudes and poleward motions at high latitudes. Several authors have suggested that these discrepancies might be ascribed to the fact that different features are anchored at different depths in the solar convection zone. Accordingly, the meridional flow deep into this zone might be reflected by the sunspot motions, whereas the meridional flow in the upper part of this zone might be reflected by the other measurements. As we shall see in Section 5.2, these speculations have a direct bearing on the theoretical models of solar differential rotation.

1.2.2 Helioseismology: The internal rotation rate

The Sun is a very small amplitude variable star. Its oscillations are arising from a huge number of discrete modes with periods ranging from a few minutes to several hours. The so-called five-minute oscillations, which have frequencies between about 2 mHz and 4 mHz, have been extensively studied. They correspond to standing acoustic waves that are trapped beneath the solar surface, with each mode traveling within a well-defined shell in the solar interior. Since the properties of these modes are determined by the stratification of the Sun, accurate measurements of their frequencies thus provide a new window in the hitherto invisible solar interior.

To a first approximation, the Sun may be considered to be a spherically symmetric body. In that case, by making use of spherical polar coordinates (r, d, y), we can write the components of the Lagrangian displacement for each acoustic mode in the separable form

where P ¡" (cos d) is the associated Legendre function of degree / and order m (—/ < m < +/). The eigenfunctions $r(r;n, /) and $h(r;n, /) define the radial and horizontal displacements of the mode. Both functions depend on the integer n, which is related to the number of zeros of the function $r along the radius, and the integer /, which is the number of nodal lines on the solar surface. Because a spherical configuration has no preferred axis of symmetry, these eigenfunctions are independent of the azimuthal order m, so that to each value of the eigenfrequency mn// correspond 21 + 1 displacements. Rotation splits this degeneracy with respect to the azimuthal order m of the eigenfrequencies. Hence, we have

Since the magnitude of the angular velocity ^ is much less than the acoustic frequencies rnnj, perturbation theory can be applied to calculate these frequency splittings. One can show that t' R rn

where the rotational kernels Knjm (r, d) are functions that may be derived from a non-rotating solar model for which one has calculated the eigenfrequencies rnnj and their corresponding eigenfunctions. Given measurements of the rotational splittings Aa)njm, it is therefore possible, in principle, to solve this integral equation for the angular velocity.

Measurement of the rotational splitting Aa)njm provides a measure of rotation in a certain region of the Sun. In fact, the acoustic modes of progressively lower I penetrate deeper into the Sun, so that the information on the angular velocity in the deeper layers is confined to splittings of low-/ modes. Similarly, because only when an acoustic mode is quasi-zonal can it reach the polar regions, the information on the angular velocity at high heliocentric latitudes is confined to splittings of low-m modes. Since the measured splittings for the low- and low-m modes have comparatively larger relative errors, determination of the function ^(r, d) thus becomes increasingly difficult with increasing depth and increasing latitude.

Several groups of workers have observed the splittings of acoustic frequencies that arise from the Sun's differential rotation. Figures 1.3 and 1.4 illustrate the inverted solution of Eq. (1.6) based on frequency splitting determinations from the latest GONG data (1996). Note that the equatorial rotation rate presents a steep increase with radius near r = 0.7R0, thus suggesting the possibility of a discontinuity near the base of the convection zone. Note also that the equatorial rotation rate peaks near r = 0.95R0, before decreasing with radius in the outermost surface layers. Figure 1.4 illustrates the latitudinal dependence of the inverted profile. In the outer convection zone, for latitude 0 < 30°, the rotation rate is nearly constant on cylinders, owing to a rapidly rotating

Fig. 1.3. Solar rotation rate inferred from the latest GONG data (1996). The curves are plotted as a function of radius at the latitudes of 0° (top), 30° (middle), and 60° (bottom). The dashed curves indicate error levels. Source: Sekii, T., in Sounding Solar and Stellar Interiors (Provost, J., and Schmider, F. X., eds.), I.A.U. Symposium No 181, p. 189, Dordrecht: Kluwer, 1997. (By permission. Copyright 1997 by Kluwer Academic Publishers.)

Fig. 1.3. Solar rotation rate inferred from the latest GONG data (1996). The curves are plotted as a function of radius at the latitudes of 0° (top), 30° (middle), and 60° (bottom). The dashed curves indicate error levels. Source: Sekii, T., in Sounding Solar and Stellar Interiors (Provost, J., and Schmider, F. X., eds.), I.A.U. Symposium No 181, p. 189, Dordrecht: Kluwer, 1997. (By permission. Copyright 1997 by Kluwer Academic Publishers.)

belt centered near r = 0.95R0. At higher latitudes, however, the rotation rate becomes constant on cones. The differential character of the rotation disappears below a depth that corresponds to the base of the convection zone. This solution agrees qualitatively with the inverted profiles obtained by other groups. Perhaps the most interesting result of these inversions is that they show no sign of a tendency for rotation to occur at constant angular velocity on cylinders throughout the outer convection zone.

In summary, several inversion studies indicate that the rotation rate in the solar convection zone is similar to that at the surface, with the polar regions rotating more slowly than the equatorial belt. Near the base of the convection zone, one finds that there exists an abrupt unresolved transition to essentially uniform rotation at a rate corresponding to some average of the rate in the convection zone. This shear layer, which is known as the solar tachocline, is centered near r = 0.7R0; recent studies indicate that it is quite thin, probably no more than 0.06R0. The actual rotation rate in the radiative core remains quite uncertain, however, because of a lack of accurately measured splittings for low-l acoustic modes. Several investigators have found that from the base of the convection zone down to r ^ 0.1-0.2R0 their measurements are consistent with uniform rotation at a rate somewhat lower than the surface equatorial rate. Not unexpectedly, the rotation rate inside that radius is even more uncertain. Some studies suggest that the rotation rate of this inner core might be between 2 and 4 times larger than that at the surface. According to other investigators, however, it is more likely that this inner core rotates with approximately the same period as the outer parts of the radiative core. I shall not go into the disputes.

Fig. 1.4. Solar rotation rate as a function of normalized radius and latitude. Contours of isorotation are shown, superimposed on a gray-scale plot of the formal errors. A very dark background means a less reliable determination. Source: Korzennik, S. G., Thompson, M. J., Toomre, J., and the GONG Internal Rotation Team, in Sounding Solar and Stellar Interiors (Provost, J., and Schmider,F. X.,eds.),I.A.U. Symposium No 181, p. 211, Dordrecht: Kluwer, 1997. (Courtesy of Dr. F. Pijpers. By permission; copyright 1997 by Kluwer Academic Publishers.)

Fig. 1.4. Solar rotation rate as a function of normalized radius and latitude. Contours of isorotation are shown, superimposed on a gray-scale plot of the formal errors. A very dark background means a less reliable determination. Source: Korzennik, S. G., Thompson, M. J., Toomre, J., and the GONG Internal Rotation Team, in Sounding Solar and Stellar Interiors (Provost, J., and Schmider,F. X.,eds.),I.A.U. Symposium No 181, p. 211, Dordrecht: Kluwer, 1997. (Courtesy of Dr. F. Pijpers. By permission; copyright 1997 by Kluwer Academic Publishers.)

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