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.     

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.     

Long-Term Variations of Solar Differential Rotation and Sunspot Activity: Revisited

Long-term variations of solar differential rotation and sunspot activity are investigated through re-analyzing the data on parameters of the differential-rotation law obtained by Makarov, Tlatov, and Callebaut (Solar Phys. 170, 373, 1997), Javaraiah, Bertello, and Ulrich (Astrophys. J. 626, 579, 2005a; Solar Phys. 232, 25, 2005b), and Javaraiah et al. (Solar Phys. 257, 61, 2009). Our results indicate that the solar-surface-rotation rate at the Equator (indicated by the A-parameter of the standard solar-rotation law) shows a secular decrease since Cycle 12 onwards, given by about 1?–?1.5×10^?3 (deg?day^?1?year^?1). The B-parameter of the standard differential-rotation law seems to also show a secular decrease since Cycle 12 onwards, but of weak statistical significance. The rotation rate averaged over latitudes 0^°?–?40^° does not show a secular trend of statistical significance. Moreover, the average sunspot area shows a secular increase of statistical significance since Cycle 12 onwards, while a negative correlation is found between the level of sunspot activity (indicated by the average sunspot area) and the solar equatorial rotation on long-term scales.     

Variations of the Solar Differential Rotation Associated with Polarity Reversal

Variations of solar differential rotation have been studied using observations of solar quiescent Hα filaments obtained during 1965–1993 at the Abastumani Astrophysical Observatory. In both hemispheres of the Sun, propagation of a quasi-biennial pulse of residual rotation velocities of filaments was found. There is a pulse drift from high latitudes to the equator in the northern hemisphere in 1968–1970, 1979–1981, 1988–1990 and in the southern one in 1969–1971, 1979–1981, 1989–1991. Propagation of a pulse starts near the time of the polarity reversal of the circumpolar regions of the Sun. High-latitude double peaks of rapid motion were found in the northern hemisphere for cycle 20 and in the southern hemisphere for cycle 22. The relation of the appearance of suggested double pulse peaks of residual velocities with the threefold polarity changing of the circumpolar areas is suggested.     

Global-Oscillation Eigenfunction Measurements of Solar Meridional Flow

We describe and apply a new helioseismic method for measuring solar subsurface axisymmetric meridional and zonal flow. The method is based on a theoretical model of the response of global-oscillation eigenfunctions to the flow velocity and uses cross spectra of the time-varying coefficients in the spherical-harmonic expansion of the photospheric Doppler-velocity field. Eigenfunction changes modify the leakage matrix, which describes the sensitivity of the spherical-harmonic coefficients to the global-oscillation modes. The form of the leakage matrix in turn affects the theoretically expected spherical-harmonic cross spectra. Estimates of internal meridional and zonal flow were obtained by fitting the theoretical flow-dependent cross spectra to spherical-harmonic cross spectra computed from approximately 500 days of full-disk Dopplergrams from the Helioseismic and Magnetic Imager (HMI) on the SDO spacecraft. The zonal-flow measurements, parameterized in the form of “a” coefficients, substantially agree with measurements obtained from conventional global-mode-frequency analysis. The meridional-flow estimates, in the form of depth-weighted averages of the flow velocity, are similar to estimates obtained from earlier analyses, for oscillation modes that penetrate the outermost one-third of the convection zone. For more deeply penetrating modes, the inferred flow velocity increases significantly with penetration depth, indicating the need for either a modification of the simple conveyor-belt picture of meridional flow or improvement in the cross-spectral model.     

Theoretical Signature of Solar Meridional Flow in Global Seismic Data

Approximate expressions are derived for the perturbations in solar p- and f-mode oscillation eigenfunctions, due to large-scale, meridional flows which are symmetric about the equator. The essential signature of the perturbed eigenfunctions in global helioseismic data is derived and the prospects for detecting meridional flow using global seismic techniques are discussed.     

Solar Variability Induced in a Dynamo Code by Realistic Meridional Circulation Variations

In this work we use an already-published method to infer a variation profile for the solar meridional circulation over the last 250 years. We feed this variation profile into a numerical dynamo code, and we reconstruct a sunspot time series that acts as a proxy for solar cycle activity. We perform three simulations with slightly different parameters, and the results are compared with the observational data. The medium and large correlation coefficients between reconstructed and observational time series seem to indicate that variations in meridional circulation play an important role in the modulation of solar activity.     

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.     

A Check on the Variations of Earth‘s Rotation with an Ancient Solar Eclipse

We address the relation between an ancient total eclipse, which occurred on A.D.1542 August 11 and the variation of Earth‘‘s rotation. The total eclipse was recorded in some ancient Chinese books, especially in local chronicles. Some of the documents include useful information for determining the location of the totality zone. The parameters of the eclipse are calculated by using the DE406 Ephemeris.A high-precision value of AT which expresses the variation of the Earth‘‘s rotation,of about 300 ～ 380 s, is obtained.     

Temporal Variations of the Solar Rotation Determined by Sunspot Groups

The extended Greenwich data set consisting of positions of sunspot groups is used for the investigation of cycle-related variations of the solar rotation in the years 1874–1981. Applying the residual method, which yields a single number for each year describing the average deviation from the mean value of the solar rotation, the dependence of the rotation velocity residual on the phase of the solar cycle is investigated. A secular deceleration of the solar rotation was found: the slope being statistically significant at the 3σ level. Periods of 33, 22, 11, 5.2, and 3.5 years can be identified in the power spectra. The rotation velocity residuals were averaged for all years with the same solar cycle phase relative to the nearest preceding sunspot minimum. The variation pattern reveals a higher than average rotation velocity in the minimum of activity and, to a lesser extent, also around the maximum of activity. The analysis was repeated with several changes in the reduction method, such as elimination of the secular trend, application of statistical weights, different cutoffs of the central meridian distance, division of the latitude into subregions and treating data from the years of activity minima separately. The results obtained are compared with those from the literature, and an interpretation of the observed phenomena is proposed.     

Iodine-Cell Spectroscopy Applied to Investigating the Solar Differential Rotation

In an attempt to examine whether the spectroscopic Doppler method with an iodine cell (which is known to be successful for precise radial-velocity determinations in stellar astronomy) could be effective for investigating the solar differential rotation, we carried out intensive observations to collect spectra at a large number of points on the solar disk by using the Domeless Solar Telescope along with the horizontal spectrograph of the Hida Observatory. Having converted the resulting line-of-sight velocity component into the angular rotational rate (ω), we derived a differential rotation law, w_sidereal  (deg day^-1) = 14.03 (±0.06)-1.84 (±0.57) sin²y-1.92 (±0.85) sin⁴y\omega_{\mathrm{sidereal}}\; (\mathrm{deg}\,\mathrm{day}^{-1}) =14.03 (\pm0.06)-1.84 (\pm0.57) \sin^{2}\psi-1.92 (\pm0.85) \sin^{4}\psi (ψ: heliographic latitude), which is reasonably consistent with other spectroscopic determinations published so far. Our analysis also revealed several practical points to note for successful application (e.g., exclusion of those data that are not well distant from the meridian; mutual data subtraction/averaging for symmetric counterparts at the eastern and western hemisphere). Considering its easiness and cheapness, this iodine-cell-featured spectroscopic method may be regarded as an effective and practical tool for studying the differential rotation of the Sun.     

Differential Rotation of the Solar Corona from Magnetic Field Data

A method for investigating the differential rotation of the solar corona using the coronal magnetic field as a tracer is proposed. The magnetic field is calculated in the potential approximation from observational data at the photospheric level. The time interval from June 24, 1976, to December 31, 2004, is considered. The magnetic field has been calculated for all latitudes from the equator to ±75? with a 5? step at distances from the base of the corona 1.0 R_⊙ to 2.45 R_⊙ near the source surface. The coronal rotation periods at 14 distances from the solar center have been determined by the method of periodogram analysis. The coronal rotation is shown to become progressively less differential with increasing heliocentric distance; it does not become rigid even near the source surface. The change in the coronal rotation periods with time is considered. At the cycleminimumthe rotation has been found to bemost differential, especially at small distances from the solar center. The change in coronal rotation with time is consistent with the tilt of the solar magnetic equator. The results from the magnetic field are compared with those obtained from the brightness of the green coronal Fe XIV 530.3 nm line. The consistency between these results confirms the reliability of the proposed method for studying the coronal rotation. Studying the rotation of the coronal magnetic field gives hope for the possibility of using this method to diagnose the differential rotation in subphotospheric layers.     

Meridional Circulation and Global Solar Oscillations

We investigate the influence of large-scale meridional circulation on solar p modes by quasi-degenerate perturbation theory, as proposed by Lavely and Ritzwoller (Roy. Soc. Lond. Phil. Trans. Ser. A 339, 431, 1992). As an input flow we use various models of stationary meridional circulation obeying the continuity equation. This flow perturbs the eigenmodes of an equilibrium model of the Sun. We derive the signatures of the meridional circulation in the frequency multiplets of solar p modes. In most cases the meridional circulation leads to negative average frequency shifts of the multiplets. Further possibly observable effects are briefly discussed.     

The Energy Lost by Differential Rotation in the Generation of the Solar Toroidal Magnetic Field

The integrals, I_i(t) = _GL u_i j × B _i dv over the volume GL are calculated in a dynamo model of the Babcock–Leighton type studied earlier. Here, GL is the generating layer for the solar toroidal magnetic field, located at the base of the solar convection zone (SCZ); i=r, , , stands for the radial, latitudinal, and azimuthal coordinates respectively; j = (4)^-1 × B, where B is the magnetic field; u_r,u are the components of the meridional motion, and u is the differential rotation. During a ten-year cycle the energy _cycle I(t)dt needs to be supplied to the azimuthal flow in the GL to compensate for the energy losses due to the Lorentz force. The calculations proceed as follows: for every time step, the maximum value of |B| in the GL is computed. If this value exceeds B_cr (a prescribed field) then there is eruption of a flux tube that rises radially, and reaches the surface at a latitude corresponding to the maximum of |B| (the time of rise is neglected). This flux tube generates a bipolar magnetic region, which is replaced by its equivalent axisymmetric configuration, a magnetic ring doublet. The erupted flux can be multiplied by a factor F_t, i.e., by the number of eruptions per time step. The model is marginally stable and the ensemble of eruptions acts as the source for the poloidal field. The arbitrary parameters B_cr and F_t are determined by matching the flux of a typical solar active region, and of the total erupted flux in a cycle, respectively. If E(B) is the energy, in the GL, of the toroidal magnetic field B = B sin cos , B (constant), then the numerical calculations show that the energy that needs to be supplied to the differential rotation during a ten-year cycle is of the order of E(B_cr), which is considerably smaller than the kinetic energy of differential rotation in the GL. Assuming that these results can be extrapolated to larger values of B_cr, magnetic fields 10⁴ G, could be generated in the upper section of the tachocline that lies below the SCZ (designated by UT). The energy required to generate these 10⁴ G fields during a cycle is of the order of the kinetic energy in the UT.     

Magnetic Field as a Tracer for Studying the Differential Rotation of the Solar Corona

The characteristics of differential rotation of the solar corona for the period 1976?–?2004 were studied as a function of the distance from the center of the Sun. For this study, we developed a method using the coronal magnetic field as a tracer. The field in a spherical layer from the base of the corona up to the source surface was determined from photospheric measurements. Calculations were performed for 14 heliocentric distances from the base of the corona up to 2.45 \(R_{\odot }\) solar radii (the vicinity of the source surface) and from the equator to \(\pm 75^{\circ }\) of latitude at \(5^{\circ }\) steps. For each day, we calculated three spherical components, which were then used to obtain the field strength. The coronal rotation periods were determined by the periodogram method. The rotation periods were calculated for all distances and latitudes under consideration. The results of these calculations make it possible to study the distribution of the rotation periods in the corona depending on distance, time, and phase of the cycle. The variations in the coronal differential rotation during the time interval 1976?–?2004 were as follows: the gradient of differential rotation decreased with the increase of heliocentric distance; the rotation remaining differential even in the vicinity of the source surface. The highest rotation rates (shortest rotation periods) were recorded at the cycle minimum at small heliospheric distances, i.e. small heights in the corona. The lowest rotation rate was observed at the middle of the ascending branch at large distances. At the minimum of the cycle, the differential rotation is most clearly pronounced, especially at small heliocentric distances. As the distance increases, the differential rotation gradient decreases in all phases. The results based on magnetic data and on the brightness of the coronal green line 530.3 nm Fe xiv used earlier show a satisfactory agreement. Since the rotation of the magnetic field at the corresponding heights in the corona is probably determined by the conditions in the field generation region, an opportunity arises to use this method for diagnostics of differential rotation in the subphotospheric layers.     

A Study of the Rotation of the Solar Corona

Synoptic photoelectric observations of the coronal Fexiv and Fex emission lines at 530.3 nm and 637.4 nm, respectively, are analyzed to study the rotational behavior of the solar corona as a function of latitude, height, time and temperature between 1976 (1983 for Fex) and 2001. An earlier similar analysis of the Fexiv data at 1.15 R over only one 11-year solar activity cycle (Sime, Fisher, and Altrock, 1989, Astrophys. J. 336, 454) found suggestions of solar-cycle variations in the differential (latitude-dependent) rotation. These results are tested over the longer epoch now available. In addition, the new Fexiv 1.15 R results are compared with those at 1.25 R and with results from the Fex line. I find that for long-term averages, both ions show a weakly-differential rotation period that may peak near 80° latitude and then decrease to the poles. However, this high-latitude peak may be due to sensing low-latitude streamers at higher latitudes. There is an indication that the Fexiv rotation period may increase with height between 40° and 70° latitude. There is also some indication that Fex may be rotating slower than Fexiv in the mid-latitude range. This could indicate that structures with lower temperatures rotate at a slower rate. As found in the earlier study, there is very good evidence for solar-cycle-related variation in the rotation of Fexiv. At latitudes up to about 60°, the rotation varies from essentially rigid (latitude-independent) near solar minimum to differential in the rising phase of the cycle at both 1.15 R and 1.25 R . At latitudes above 60°, the rotation at 1.15 R appears to be nearly rigid in the rising phase and strongly differential near solar minimum, almost exactly out of phase with the low-latitude variation.     

Rotation of the Solar Equator

Regular measurements of the general magnetic field of the Sun, performed over about half a century at the Crimean Astrophysical Observatory, the J. Wilcox Solar Observatory, and five other observatories, are considered in detail for the time 1968?–?2016. They include more than twenty-six thousand daily values of the mean line-of-sight field strength of the visible solar hemisphere. On the basis of these values, the equatorial rotation period of the Sun is found to be 26.926(9) d (synodic). It is shown that its half-value coincides within error limits with both the main period of the magnetic four-sector structure, 13.4577(25) d, and the best-commensurate period of the slow motions of the major solar system bodies, 13.479(22) d (sidereal). The probability that the two periods coincide by chance is estimated to be about \(10^{-7}\). The true origin of this odd resonance is unknown.     

Solar Rotation in the 17th century

Sunspot drawings made by Galileo Galilei in 1612 are used to derive the law of differential rotation at that time. The main interest of the work is during the time of observations, just at the beginning of telescopic observations and some decades before the Maunder Minimum (1645 – 1715), a period where the sunspots almost disappeared from the solar surface. For this purpose we have carried out careful corrections of the different sources of errors derived from the observing technique. By comparing with other results of the same century, a significant difference is only detected by comparing with data corresponding to the deep Maunder Minimum (Paris Observatory drawings). The characteristics of the solar differential rotation, and extrapolating the behavior of solar activity, did not differ before or after the Maunder Minimum. We also include an analysis of hitherto ignored sunspot drawings by N. Bion made in October and November 1672.     

A Measure of the Solar Rotation During the Maunder Minimum

In this paper we present a measure of the synodic solar rotation rate derived from an analysis of a Flamsteed drawing, corroborating the decrease of the solar rotation in the deep Maunder minimum (1666–1700).