Astron. Astrophys. 321, 323-329 (1997)

## 5. Reynolds stresses

In this section, we address the important question of the angular momentum transport, which is related to the origin of the differential rotation. Let us use u and v to denote the latitudinal and longitudinal motions, respectively. The m.c. u is now positive when equatorward in both hemispheres. The transport of linear angular momentum across a given latitude circle (Ward 1965) is proportional to

where [ ] indicate spatial averages and spatial deviations from this average; indicate time averages and [ ] are deviations from the time-space average. The first term, , represents the transport by the meridional flow (hereafter "M.C.T."). The second and third terms represent the transport by non-axisymmetrical eddies. We cannot separate these but they measure the correlation between the two horizontal motions

This correlation (or "covariance") term is a measure of the latitudinal angular momentum transport via Reynolds stresses. Positive covariance implies that higher-than-average longitudinal motion in the direction of rotation is coupled with equatorward motion (see Paper II for more details).

### 5.1. Covariance of cycle 19 sunspots

The covariance has been calculated within each latitude bin, and for all cycle 19 Meudon sunspots (Table 4). The covariance is found to be very small and mainly positive, in the northern hemisphere, while negative in the southern. Fig. 6 shows the covariance obtained for all cycle 19 sunspots. The two hemispheres have been folded. We observe a very small covariance (less than 0.02 (deg/day)2). As previously noted in Sect. 4, there seems to be a very small contribution to the equatorial acceleration at low latitudes and to poleward acceleration at high latitudes. Our results are in agreement with those for cycle 21 sunspots (Paper II), unlike those of the Mount Wilson analysis (Gilman & Howard 1984) and Greenwich sunspot groups (Ward 1965).

Table 4. Angular momentum transport by Reynolds stresses and by meridional flow, for the whole period (1957-1962). Positive covariance implies that higher-than-average longitudinal motion in the direction of rotation is coupled with equatorward motion

 Fig. 6. Sunspot covariance during cycle 19 (Meudon, solid line), cycle 21 (Meudon, dashed line), Mount Wilson sunspots (1921-1982, dot-dashed line) and Greenwich sunspots (1925-1954, dotted line)

We also compare the covariance between cycle 19 and 21 sunspots for each hemisphere, separately, in Fig. 5.1. In each case, the covariance is of the same order of magnitude. The observed differences between the two hemispheres are within the noise level.

 Fig. 7. Meudon sunspot covariance during cycles 19 (solid line) and 21 (dashed line). Left: northern hemisphere. Right: southern hemisphere

### 5.2. Leading and following sunspot covariance

Considering now whether the covariance varies with sunspot polarity and age, we take the covariance for leading and following sunspots shown in Fig. 8. The two hemispheres have been folded. Leaders seem to exhibit a higher covariance than followers (positive for leaders and negative for followers). This is probably due to the tilt angle of the emerging bipolar group.

 Fig. 8. Covariance of leading () and following () sunspots during cycles 19 (1957-1962, solid line) and 21 (1977-1984, dashed line) from Meudon data

### 5.3. Old/young sunspot covariance

Fig. 9 displays young and old sunspot covariances for cycle 19. We observed no strong difference between the two sunspot classes. This does not confirm the higher covariance for young sunspots as detected in Paper II. The difference in covariance between leading and following sunspots and between old and young sunspots is probably not significant. The robust result is that sunspots do not show much significant transport via Reynolds stresses.

 Fig. 9. Covariance of young () and old () sunspots during cycles 19 (1957-1962, solid line) and 21 (1977-1984, dashed line: young sunspots and dotted line: old sunspots) from Meudon data

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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