Long-Term Variations in the Solar Differential Rotation

Using Greenwich data (1879–1976) and SOON/NOAA data (1977–2002) on sunspot groups we found the following results: (i) The Sun's mean (over all the concerned cycles during 1879–1975) equatorial rotation rate (A) is significantly larger (≈0.1%) in the odd-numbered sunspot cycles (ONSCs) than in the even-numbered sunspot cycles (ENSCs). The mean rotation is significantly (≈10%) more differential in the ONSCs than in the ENSCs. North–south difference in the mean equatorial rotation rate is larger in the ONSCs than in the ENSCs. North–south difference in the mean latitude gradient of the rotation is significant in the ENSCs and insignificant in the ONSCs. (ii) The known very large decrease in A from cycle 13 to cycle 14 is confirmed. The amount of this decrease in the mean A was about 0.017 μrad s⁻¹. Also, we find that A decreased from cycle 17 to cycle 18 by about 0.008 μrad s⁻¹ and from cycle 21 to cycle 22 by about 0.016 μrad s⁻¹. From cycle 13 to cycle 14 the decrease in A was more in the northern hemisphere than in the southern hemisphere, it is opposite in the later two epochs. The time gap between the consecutive drops in A is about 44 years, suggesting the existence of a `44-yr' cycle or `double Hale cycle' in A. The time gap between the two large drops, viz., from cycle 13 to cycle 14 and from cycle 21 to cycle 22, is about 90 years (Gleissberg cycle). We predict that the next drop (moderate) in A will be occurring from cycle 25 to cycle 26 and will be followed by a relatively large-amplitude `double Hale cycle' of sunspot activity. (iii) Existence of a 90-yr cycle is seen in the cycle-to-cycle variation of the latitude gradient (B). A weak 22-yr modulation in B seems to be superposed on the relatively strong 90-yr modulation. (iv) The coefficient A varies significantly only during ONSCs and the variation has maximum amplitude in the order of 0.01 μrad s<sup>−1</sup> around activity minima. (v) There exists a good anticorrelation between the mean variation of B during the ONSCs and that during the ENSCs, suggesting the existence of a `22-yr' periodicity in B. The maximum amplitude of the variation of B is of the order of 0.05 μrad s⁻¹ around the activity minima. (vi) It seems that the well-known Gnevyshev and Ohl rule of solar activity is applicable also to the cycle-to-cycle amplitude modulation of B from cycle 13 to cycle 20, but the cycles 12 (in the northern hemisphere, Greenwich data) and 21 (in both hemispheres, SOON/NOAA data) seem to violate this rule in B. And (vii) All the aforesaid statistically significant variations in A and B seem to be related to the approximate 179-yr cycle, 1811–1989, of variation in the Sun's motion about the center of mass of the solar system.     

Search for Short-Term Periodicities in the Sun’s Surface Rotation: A Revisit

We have used the daily values of the equatorial rotation rate determined from the Mt. Wilson daily Doppler-velocity measurements during the period 3 December 1985 – 5 March 2007 to search for periodicities in the solar equatorial rotation rate on time scales shorter than 11 years. After the daily values have been binned into 61-day intervals, a cosine fit with a period of one year was applied to the sequence to remove any seasonal trend. The spectral properties of this sequence were then investigated by using standard Fourier analysis, maximum-entropy methods, and a Morlet-wavelet analysis. From the analysis of the Fourier power spectrum we detected peaks with periodicities around 7.6, 2.8, and 1.47 years and 245, 182, and 158 days, but none of them were at a statistically significant level. In the Morlet-wavelet analysis the ≈1.47-year periodicity is detected only for 1990 (i.e., near the maximum of cycle 22) and near the end of cycle 22 in 1995. From the same wavelet analysis we found some evidence for the existence of a 2.8-year periodicity and a 245-day periodicity in the equatorial rotation rate around the years 1990 and 1992, respectively. In the data taken during the period 1996 – 2007, when the Mt. Wilson spectrograph instrumentation was more stable, we were unable to detect any signal from the wavelet analysis. Thus, the detected periodicities during the period before 1996 could be artifacts of frequent changes in the Mt. Wilson spectrograph instrumentation. However, the temporal behavior of most of the activity phenomena during cycles 22 (1986 – 1996) and 23 (after 1997) is considerably different. Therefore, the presence of the aforementioned short-term periodicities during the last cycle and absence of them in the current cycle may, in principle, be real temporal behavior of the solar rotation during these cycles.     

Long-Term Variations in Solar Differential Rotation and Sunspot Activity

The solar equatorial rotation rate, determined from sunspot group data during the period 1879–2004, decreased over the last century, whereas the level of activity has increased considerably. The latitude gradient term of the solar rotation shows a significant modulation of about 79 year, which is consistent with what is expected for the existence of the Gleissberg cycle. Our analysis indicates that the level of activity will remain almost the same as the present cycle during the next few solar cycles (i.e., during the current double Hale cycle), while the length of the next double Hale cycle in sunspot activity is predicted to be longer than the current one. We find evidence for the existence of a weak linear relationship between the equatorial rotation rate and the length of sunspot cycle. Finally, we find that the length of the current cycle will be as short as that of cycle 22, indicating that the present Hale cycle may be a combination of two shorter cycles. Presently working for the Mt. Wilson Solar Archive Digitization Project at UCLA.     

Study of Distribution and Asymmetry of Solar Active Prominences during Solar Cycle 23

In this article we present the results of a study of the spatial distribution and asymmetry of solar active prominences (SAP) for the period 1996 through 2007 (solar cycle 23). For more meaningful statistical analysis we analyzed the distribution and asymmetry of SAP in two subdivisions viz. Group1 (ADF, APR, DSF, CRN, CAP) and Group2 (AFS, ASR, BSD, BSL, DSD, SPY, LPS). The North – South (N – S) latitudinal distribution shows that the SAP events are most prolific in the 21° to 30° slice in the Northern and Southern Hemispheres; the East – West (E – W) longitudinal distribution study shows that the SAP events are most prolific (best observable) in the 81° to 90° slice in the Eastern and Western Hemispheres. It was found that the SAP activity during this cycle is low compared to previous solar cycles. The present study indicates that during the rising phase of the cycle the number of SAP events are roughly equal in the Northern and Southern Hemispheres. However, activity in the Southern Hemisphere has been dominant since 1999. Our statistical study shows that the N – S asymmetry is more significant then the E – W asymmetry.     

Solar differential rotation derived from sunspot observations

Sunspot drawings obtained at National Astronomical Observatory of Japan during the years 1954–1986 were used to determine the differential rotation of the Sun. From the limited data set of three solar cycles it was found that three factors (the level of cycle activity, the cycle phase, and sunspot type) affect the solar rotation rate. The differential rotation varies from cycle to cycle in such a way that the rotation velocity in the low activity cycle (cycle 20) is higher than in the high-activity cycle (cycle 19). The equatorial rotation rate shows a systematic variation within each cycle. The rate is higher at the beginning of the cycle and decreases subsequently. Although quite small, the variation of solar differential rotation with respect to Zürich sunspot type was found. The H and J types show the slowest rotation among all the sunspot types.     

On the Long-Term Modulation of Solar Differential Rotation

Long-term modulation of solar differential rotation was studied with data from Mt. Wilson and our original observations during Solar Cycles 16 through 23. The results are that i) the global B-value (i.e. latitudinal gradient of differential rotation) is modulated with a period of about six or seven solar cycles, ii) the B-values of the northern and southern hemispheres are also modulated with a period similar to the global one, but iii) they show quasi-oscillatory behavior with a phase shift between them. We examined the yearly fluctuations of the B-values in every solar cycle with reference to the phase of the sunspot cycle and found that the B-values in the sunspot-minimum years show large and erratic variations, while those in the sunspot-maximum years show small fluctuations. Positive correlation between the former B-values and the latter was found. We discuss the independent long-term behavior of solar differential rotation between the northern and southern solar hemispheres and the implication for the solar dynamo.     

The Rotation Rates of Solar Magnetic Fields During Solar Cycles 21 – 23

Employing the synoptic maps of the photospheric magnetic fields from the beginning of solar cycle 21 to the end of 23, we first build up a time – longitude stackplot at each latitude between ±35°. On each stackplot there are many tilted magnetic structures clearly reflecting the rotation rates, and we adopt a cross-correlation technique to explore the rotation rates from these tilted structures. Our new method avoids artificially choosing magnetic tracers, and it is convenient for investigating the rotation rates of the positive and negative fields by omitting one kind of field on the stackplots. We have obtained the following results. i) The rotation rates of the positive and negative fields (or the leader and follower polarities, depending on the hemispheres and solar cycles) between latitudes ±35° during solar cycles 21–23 are derived. The reversal times of the leader and follower polarities are usually not consistent with the years of the solar minimum, nevertheless, at latitudes ±16°, the reversal times are almost simultaneous with them. ii) The rotation rates of the three solar cycles averaged over each cycle are calculated separately for the positive, negative and total fields. The latitude profiles of rotation of the positive and negative fields exhibit equatorial symmetries with each other, and those of the total fields lie between them. iii) The differences in rotation rates between the leader and follower polarities are obtained. They are very small near the equator, and increase as latitude increases. In the latitude range of 5° – 20°, these differences reach 0.05 deg day⁻¹, and the mean difference for solar cycle 22 is somewhat smaller than cycles 21 and 23 in these latitude regions. Then, the differences reduce again at latitudes higher than 20°.     

Determination of the Rotation Periods of Solar Active Longitudes

There are two types of active longitudes (ALs) in terms of the distribution of sunspot areas: long-lived and intra-cyclic ALs. The rotation period of the long-lived ALs has been determined by a new method in this paper. The method is based on the property of ALs to be maintained over several cycles of solar activity. The daily values of sunspot areas for 1878 – 2005 are analyzed. It is shown that the AL positions remain almost constant over a period of about ten cycles, from cycle 13 to cycle 22. The rotation period was found to be 27.965 days during this period. The dispersion in AL positions is about 26° from cycle to cycle, which is half of the dispersion observed in the Carrington system. The ALs in the growth phase of the activity cycle are more stable and pronounced. The excess in solar activity in the ALs over adjacent longitudinal intervals is about 12 – 14%. It is shown that only one long-lived AL can be observed at one time on the Sun, as a rule.     

Gnevyshev Peaks and Gaps for Coronal Mass Ejections of Different Widths Originating in Different Solar Position Angles

The sunspot number series at the peak of sunspot activity often has two or three peaks (Gnevyshev peaks; Gnevyshev, Solar Phys. 1, 107, 1967; Solar Phys. 51, 175, 1977). The sunspot group number (SGN) data were examined for 1997 – 2003 (part of cycle 23) and compared with data for coronal mass ejection (CME) events. It was noticed that they exhibited mostly two Gnevyshev peaks in each of the four latitude belts 0° – 10°, 10° – 20°, 20 ° – 30°, and > 30°, in both N (northern) and S (southern) solar hemispheres. The SGN were confined to within latitudes ± 50° around the Equator, mostly around ± 35°, and seemed to occur later in lower latitudes, indicating possible latitudinal migration as in the Maunder butterfly diagrams. In CMEs, less energetic CMEs (of widths < 71°) showed prominent Gnevyshev peaks during sunspot maximum years in almost all latitude belts, including near the poles. The CME activity lasted longer than the SGN activity. However, the CME peaks did not match the SGN peaks and were almost simultaneous at different latitudes, indicating no latitudinal migration. In energetic CMEs including halo CMEs, the Gnevyshev peaks were obscure and ill-defined. The solar polar magnetic fields show polarity reversal during sunspot maximum years, first at the North Pole and, a few months later, at the South Pole. However, the CME peaks and gaps did not match with the magnetic field reversal times, preceding them by several months, rendering any cause – effect relationship doubtful.     

Starspots, Activity Cycles, and Differential Rotation on Cool Stars

We present the first results of searching for stellar cycles by analysis of stellar spottedness using an algorithm developed at the Crimean Astrophysical Observatory. For more than 35 red spotted stars, we find ten targets which demonstrate cyclic variations of average latitudes and total areas of starspots. Activity cycles detected by this method have a typical cycle length about 4–15 years which are analogous to the 11-year solar Schwabe cycle. Most of the program stars demonstrate a rough analogue with the solar butterfly diagram. They show a tendency for the average starspot latitude lowering when the total spot area grows. At the same time these stars show variations of stellar photometric period (which is traced by starspots) with the starspot latitudinal drift analogously to the solar differential rotation effect. We suspect that the starspot latitudinal drift rate and the differential rotation gradient depend on the stellar spectral type.     

Investigation of the Solar Differential Rotation of Compact Magnetic Elements for 1966 – 1986

The differential rotation of compact magnetic elements during activity cycles 20 and 21 (1966 – 1986) is studied by using solar synoptic charts. For each hemisphere the compact magnetic elements with the polarity of the circumpolar magnetic field have larger rotation rates than the elements with the opposite polarity. This difference in rotation rates is present during the whole cycle except during the polarity reversal of the circumpolar field.     

The 100—year periodic modulation of solar rotation

A periodic long-term modulation of the solar surface rotation with a time scale on the order of 100 years is found in the sunspot data from 1874 to 1992 obtained by combinig the Greenwich Photoheliographic Results from cycle 11 to cycle 20 analysed by Balthasar, Vázquez, and Wöhl and the Mitaka sunspot sketch data from cycle 18 to 22 of the National Astronomical Observatory of Japan which was the Tokyo Astronomical Observatory of the University of Tokyo until 1988. A new index of the solar rotation M defined by integrating the angular momentum density over the whole surface, which we call the angular momentum surface layer density, reached a maximum at solar cycle 14, decreased to a minimum at cycle 17, and then increased to reach another maximum at cycle 21. The increase of M means acceleration of the surface layer as a whole by transport of angular momentum from the deeper layer. This implies an decrease (increase) of the radial gradient of the differential rotation if the basic radial gradient of the differential rotation increases (decreaes) inward. The decrease of M means deceleration of the surface layer and implies an increase (decrease) of the radial gradient. The degree of the equatorial acceleration of the surface differential rotation is also found to have undergone the same 100 year periodic modulation during the same interval, reaching a minimum at cycle 14, a maximum at cycle 17, and a minimum at cycle 21 in antiphase with the modulation of M. Thus both radial and latitudinal gradients of the differential rotation increased and decreased in phase (in anti-phase) if the basic radial gradient increases (decreases) inward.     

A Prediction of Solar Cycle 24 Using a Modified Precursor Method

Based on cycles 17 – 23, linear correlations are obtained between 12-month moving averages of the number of disturbed days when Ap is greater than or equal to 25, called the Disturbance Index (DI), at thirteen selected times (called variate blocks 1, 2,… , each of six-month duration) during the declining portion of the ongoing sunspot cycle and the maximum amplitude of the following sunspot cycle. In particular, variate block 9, which occurs just prior to subsequent cycle minimum, gives the best correlation (0.94) with a minimum standard error of estimation of ± 13, and hindcasting shows agreement between predicted and observed maximum amplitudes to about 10%. As applied to cycle 24, the modified precursor technique yields maximum amplitude of about 124±23 occurring about 45±4 months after its minimum amplitude occurrence, probably in mid to late 2011.     

Differential Rotation of Strong Magnetic Flux During Solar Cycles 21 – 23

In the present investigation we measure the differential rotation of strong magnetic flux during solar cycles 21 – 23 with the method of wavelet transforms. We find that the cycle-averaged synodic rotation rate of strong magnetic flux can be written as ω=13.47−2.58sin ² θ or ω=13.45−2.06sin ² θ−1.37sin ⁴ θ, where θ is the latitude. They agree well with the results derived from sunspots. A north–south asymmetry of the rotation rate is found at high latitudes (28°<θ<40°). The strong flux in the southern hemisphere rotates faster than that in the northern hemisphere by 0.2 deg day⁻¹. The asymmetry continued for cycles 21 – 23 and may be a secular property.     

Relationship between the Parameters of Coronal Holes and High-Speed Solar Wind Streams over an Activity Cycle

The comparison of the brightness and area of coronal holes (CH) to the solar wind speed, which was started by Obridko et al. (Solar Phys. 260, 191, 2009a) has been continued. While the previous work was dealing with a relatively short time interval 2000 – 2006, here we have analyzed the data on coronal holes observed in the Sun throughout activity Cycle 23. A catalog of equatorial coronal holes has been compiled, and their brightness and area variations during the cycle have been analyzed. It is shown that CH is not merely an undisturbed zone between the active regions. The corona heating mechanism in CH seems to be essentially the same as in the regions of higher activity. The reduced brightness is the result of a specific structure with the magnetic field being quasi-radial at as low an altitude as 1.1R _⊙ or a bit higher. The plasma outflow decreases the measure of emission from CH. With an adequate choice of the photometric boundaries, the CH area and brightness indices display a fairly high correlation (0.6 – 0.8) with the solar wind velocity throughout the cycle, except for two years, which deviate dramatically – 2001 and 2007, i.e., the maximum and the minimum of the cycle. The mean brightness of the darkest part of CH, where the field lines are nearly radial at low altitudes, is of the order of 18 – 20% of the solar brightness, while the brightness of the other parts of the CH is 30 – 40%. The solar wind streams originate at the base of the coronal hole, which acts as an ejecting nozzle. The solar wind parameters in CH are determined at the level where the field lines are radial.     

Prediction of Solar Cycle Maximum Using Solar Cycle Lengths

If the rise time RT, fall time FT, and total time TT (i.e., RT+FT) of a solar cycle are compared against the maximum amplitude Rz(max ) for the following cycle, then only the association between TT and Rz(max ) is inferred to be well anticorrelated, inferring that the larger (smaller) the value of Rz(max ) for the following cycle, the shorter (longer) the TT of the preceding cycle. Although the inferred correlation (−0.68) is statistically significant, the inferred standard error of estimate is quite large, so predictions using the inferred correlation are not very precise. Removal of cycle pairs 15/16, 19/20, and 20/21 (statistical outliers) yields a regression that is highly statistically significant (−0.85) and reduces the standard error of estimate by 18%. On the basis of the adjusted regression and presuming TT=140 months for cycle 23, the present ongoing cycle, cycle 24’s 90% prediction interval for Rz(max ) is estimated to be about 94±44, inferring only a 5% probability that its Rz(max ) will be larger than about 140, unless of course cycle pair 23/24 is a statistical outlier.     

The Rotation of Sunspots During Activity Cycle 22

The solar rotation rate derived from the measurements of daily sunspot positions during activity cycle 22 (1988–1996) are presented. The averaged differential rotation is nearly symmetric with respect to the equator. The comparisons of our measurement with those for previous cycles show that the equatorial rotation rate gradually increases from cycle 18 to 22.     

Periodic Appearance of Coronal Holes and the Related Variation of Solar Wind Parameters

We compared the variability of coronal hole (CH) areas (determined from daily GOES/SXI images) with solar wind (daily ACE data) and geomagnetic parameters for the time span 25 January 2005 until 11 September 2005 (late declining phase of solar cycle 23). Applying wavelet spectral analysis, a clear 9-day period is found in the CH time series. The GOES/SXI image sequence suggests that this periodic variation is caused by a mutual triangular distribution of CHs ∼120° apart in longitude. From solar wind parameters a 9-day periodicity was obtained as well, simultaneously with the 9-day period in the CH area time series. These findings provide strong evidence that the 9-day period in solar wind parameters, showing up as higher harmonic of the solar rotation frequency, is caused by the “periodic” longitudinal distribution of CHs on the Sun recurring for several solar rotations. The shape of the wavelet spectrum from the Dst index matches only weakly with that from the CH areas and is more similar to the wavelet spectrum of the solar wind magnetic field magnitude. The distinct 9-day period does not show up in sunspot group areas which gives further evidence that the solar wind modulation is strongly related to CH areas but not to active region complexes. The wavelet power spectra for the whole ACE data range (∼1998 – 2006) suggest that the 9-day period is not a singular phenomenon occurring only during a specific time range close to solar minimum but is occasionally also present during the maximum and decay phase of solar cycle 23. The main periods correspond to the solar rotation (27^d) as well as to the second (13.5^d) and third (9^d) harmonic. Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users.     

Solar Cycle Predictions Based on Extrapolation of Spectral Components: An Update

Three series (1876 – 1986, 1886 – 1996, and 1896 – 2006) of 111 annual values of sunspot number R _z in each were subjected to spectral analysis to detect periodicities by the maximum entropy method (MEM), and the periodicities so obtained were used in a multiple regression analysis (MRA) to estimate the amplitudes and phases. All series showed roughly similar spectra with many periodicities (24 or more), but most of these were insignificant. The significant periodicities (far exceeding 2σ) were near 5, 8 – 12, 18, and 37 years. Using the amplitudes and phases of these, we obtained reconstructed series, which showed good correlations (+ 0.7 and more) with the original series. When extrapolated further in time, the reconstructed series indicated R _z(max) in the ranges 80 – 101 (mean 92) for cycle 24 during years 2011 – 2014, 112 – 127 (mean 119) for cycle 25 during years 2022 – 2023, 115 – 120 (mean 118) for cycle 26 during years 2032 – 2034, and 100 – 113 (mean 109) for cycle 27 during 2043 – 2045.     

Solar rotation in the chromosphere and corona

Extreme ultraviolet spectroheliograms in Mg x (625 Å) and the Lyman continuum (897 Å) obtained from OSO-6 are used to determine the differential rotation rate in the solar chromosphere and corona. The equatorial rotation rate agrees with spectroscopic measurements of the photospheric plasma velocity; the variation of rate with latitude is less pronounced than in most other determinations. We cannot discern a variation in the rotation rate between the chromosphere and corona.