ALBERT

Notes: Abstract Using Greenwich data on sunspot groups during 1874–1976, we have studied the temporal variations in the differential rotation parametersA andB by determining their values during moving time intervals of lengths 1–5 yr successively displaced by 1 yr. FFT analysis of the temporal variations ofB (orB/A) shows periodicities 18.3 ± 3 yr, 8.5 ± 1 yr, 3.9 ± 0.5 yr, 3.1 ± 0.2 yr, and 2.6 ± 0.2 yr at levels ≥ 2σ. This analysis also shows five more periodicities at levels 1–2σ. The maximum entropy method is used to set narrower limits on the values of these periods. The reality of the existence of all these periodicities ofB (orB/A ) except the one at 2.8 yr is confirmed by analyzing the simulated time series ofB andB/A with values ofA andB randomly distributed within the limits of their respective uncertainties. Four of the prominent periods ofB agree, within their uncertainties, with the known periods in the the large-scale photospheric magnetic field. The deviations from the average differential rotation are larger near the sunspot minima. On longer time scales, the variations in the amount of sunspot activity per unit time are well correlated to the variations in the amplitudes of the ‘torsional oscillation’ represented by the 22-yr periodicity inB. All the periods inB found here are in good agreement with the synodic periods of two or more consecutive planets. The possibility of planetary configurations providing perturbations needed for the Sun's MHD torsional oscillations is speculated upon and briefly discussed.

Notes: Abstract The ‘sunspot occurrence probability’ defined in Paper I is used to determine the Legendre-Fourier (LF) terms in the ‘rate of emergence of toroidal magnetic flux,Q(θ, t), above the photosphere per unit latitude interval, per unit time’. Assuming that the magnetic flux tubes whose emergence yields solar activity are produced by interference of global MHD waves in the Sun, we determine how the amplitudes and phases of the LF terms in the toroidal magnetic fieldB Φ, representing the waves, will be related to those of the LF terms inQ(θ, t). The set of LF terms in ‘Q’ that represents the set of waves whose interference produces most of the observed sunspot activity is {l = 1, 3, ⋯, 13;v =nv *,n = 1, 3, 5}, wherev * = 1/21.4 yr−1. However, among the ‘shapes’ of sunspot cycles modeled using various sets of the computed LF terms the best agreement with the observed shape, for each cycle, is given by the set {l = 3 orl = 3, 5; andn = 1, 3 orn = 1, 3, 5}. The sets of terms: {l = 1, 3, 5, 7;n = 1}, {l = 1, 3, 5, 7;n = 3}, {l = 9, 11, 13, 15;n = 1} and {l = 9, 11, 13, 15;n = 3} seem to represent four modes of global MHD oscillation. Correlations between the amplitudes (and phases) of LF terms in different modes suggest possible existence of cascade of energy from constituent MHD waves of lowerl andn to those of higherl andn. The spectrum of the MHD waves trapped in the Sun may be maintained by the combined effect of this energy cascade and the loss of energy in the form of the emerging flux tubes. The primary energy input into the spectrum may be occurring in the mode {l = 1, 3, 5, 7;n = 1). As expected from the above phenomenological model, the size of a sunspot cycle and its excess over the previous cycle are well correlated (e.g., ∼ 90%) to the phase-changes of the two most dominant oscillation modes during the previous one or two cycles. These correlations may provide a physical basis to forecast the cycle sizes.

Notes: Abstract We have studied the temporal variations in the north–south asymmetries of the differential rotation parameters A, B, and the ‘mean’ rotation rate Ā, by determining their values from Greenwich data for sunspot groups (1879–1976) in the northern and southern hemispheres, during moving time intervals of lengths 3 yr and 5 yr, successively displaced by 1 yr. The variation in the north–south asymmetry (Ā $$a $$ ) of Ā is similar to the variation in the asymmetry (B $$a $$ ) of B but with opposite sign. These variations of Ā $$a $$ and B $$a $$ may represent components of an anti-symmetric torsional oscillation which are in opposite phase with each other. The FFT and MEM analyses of the temporal variations of B $$a $$ , Ā $$a $$ , and the north–south asymmetry (A $$a $$ ) of A, show existence of significant periodicities: 45.5 ± 11.5 yr,21.3 ± 4.5 yr, 13.3 ± 1.5 yr, and 10.5 ± 0.5 yr. These analyses also show a few other possible periodicities in A $$a $$ , B $$a $$ , and Ā $$a $$ . All these periodicities are also seen in the north–south asymmetry of sunspot activity (with similar relative magnitudes). The 22-yr periodicity was seen in ‘even-parity’ modes of magnetic field inferred from sunspot data.

Notes: Abstract We have looked for periodicities in solar differential rotation on time scales shorter than the 11-year solar cycle through the power- spectrum analysis of the differential rotation parameters determined from Mt. Wilson velocity data (1969–1994) and Greenwich sunspot group data (1879–1976). We represent the differential rotation by a set of Gegenbauer polynomials (ω(φ)= $$\bar A$$ + $$\bar B$$ (5sin2φ−1)+ $$\bar C$$ (21sin4φ−14sin2φ+1)). For the Mt. Wilson data, we focus on observations obtained after 1981 due to the reduced instrumental noise and have binned the data into intervals of 19 days. We calculated annual averages for the sunspot data to reduce the uncertainty and corrected for outliers occuring during solar cycle minima. The power spectrum of the photospheric ‘mean rotation’ $$\bar A$$ , determined from the velocity data during 1982–1994, shows peaks at the periods of 6.7–4.4 yr, 2.2 ± 0.4 yr, 1.2 ± 0.2 yr, and 243 ± 10 day with ≥99.9% confidence level, which are similar to periods found in other indicators of solar activity suggesting that they are of solar origin. However, this result has to be confirmed with other techniques and longer data sets. The 11-yr periodicity is insignificant or absent in $$\bar A$$ . The power spectra of the differential rotation parameters $$\bar B$$ and $$\bar C$$ , determined from the same subset, show only the solar cycle period with a ≥99.9% confidence level. The time series of $$\bar A$$ determined from the yearly sunspot group data obtained during 1879–1976 is very similar to the corresponding time series of $$\bar B$$ . After correcting for data with large error bars (occurring during cycle minima), we find periods, which are most likely harmonics of the solar cycle, such as 18.3 ± 3.0 yr and 7.5 ± 0.5 yr in $$\bar A$$ and confirmed these and the 3.0 ± 0.1 yr period in . The original time series show in addition some shorter periods, absent in the corrected data, representing temporal variations during cycle minimum. Given their large error bars, it is uncertain whether they represent a solar variation or not. The results presented here show considerable differences in the periodicities of $$\bar A$$ and $$\bar B$$ determined from the velocity data and the spot group data. These differences may be explained by assuming that the rotation rates determined from velocity and sunspot data represent the rotation rates of the Sun's surface layers and of somewhat deeper layers.

Notes: Abstract We have analyzed data on sunspot groups compiled during 1874–1981 and investigated the following: (i) dependence of the `initial' meridional motion (v ini(τ)) of sunspot groups on the life span (τ) of the groups in the range 2–12 days, (ii) dependence of the meridional motion (v(t)) of sunspot groups of life spans 10–12 days on the age (t) of the spot groups, and (iii) variations in the mean meridional motion of spot groups of life span 2–12 days during the solar cycle. In each of the latitude intervals 0°–10°, 10°–20° and 20°–30°, the values of both v ini(τ) and v(t) often differ significantly from zero. In the latitude interval 20°–30°, the forms of v ini(τ) and v(t) are largely systematic and mutually similar in both the north and south hemispheres. The form of v(t) suggests existence of periodic variation in the solar meridional motion with period of 4 days and amplitude 10–20 m s−1. Using the anchoring depths of magnetic structures for spot groups of different τ and testimated earlier, (Javaraiah and Gokhale, 1997), we suggest that the forms of v ini(τ) and v(t) may represent radial variation of meridional flow in the Sun's convection zone, rather than temporal variation of the flow. The meridional flows (v e(t)) determined from the data during the last few days (i.e., age t: 10–12 days) of spot groups of life spans of 10–12 days are found to have magnitudes (10–20 m s−1) and directions (poleward) similar to the those of the surface meridional plasma flows determined from the Dopplergrams and magnetograms. The mean meridional velocity of sunspot groups living 2–12 days seems to vary during the solar cycle. The velocity is not significantly different from zero during the rising phase of the cycle and there is a suggestion of equatorward motion (a few m s−1at lower latitudes and ∼10 m s−1at higher latitudes) during the declining phase (last few years) of the cycle. The variation during the odd numbered cycles seems to anticorrelate with the variation during the even numbered cycles, suggesting existence of ∼22-year periodicity in the solar meridional flow. The amplitude of the anticorrelation seems to be depending on latitude and the cycle phase. In the latitude interval 20°–30° the `surface plasma meridional motion', v e(t), is found to be poleward during maximum years (v e(t) ∼20 m s−1at 4th year) and equatorward during ending years of the cycle (v e(t) ∼−17 m s−1at 10th year).

Notes: Abstract The spherical-harmonic-Fourier analysis of the Sun's magnetic field inferred from the Greenwich sunspot data is refined and extended to include the full length (1874–1976) of the data on the magnetic tape provided by H. Balthasar. Perspective plots and grey level diagrams of the SHF power spectra for the odd and the even degree axisymmetric modes are presented. Comparing these with spectra obtained from two simulated data sets with random redistribution within the wings in the butterfly diagrams, we conclude that there is no clear evidence for the existence of any relation between the harmonic degree and the temporal frequency of the power concentrations of the inferred field. Apart from the power ‘ridge’ in the narrow frequency band at ∼ 1/21.4 y −1, and low ridges at odd multiples of this frequency, there are no other spectral features. This strongly suggests that the solar magnetic cycle consists of some global oscillations of the Sun ‘forced’ at a frequency ∼ 1/21.4 y −1 and, perhaps, weak resonances at its odd harmonics. The band width of the forcing frequency seems to be much less than 1/107 y −1. In case the global oscillations are torsional MHD, the significance of their parity and power peak is pointed out.

Notes: Abstract We show that the axisymmetric odd degree SHF modes of 21.4-yr periodicity and degrees l ≤ 29 in the solar magnetic field (as inferred from sunspot data during 1874–1976), are at least approximately stationary. Among the sine and cosine components of these SHF modes we find four groups, each defining the geometry of a coherent global oscillation characterized by a distinct power hump and its own level of variation. The first two of these ‘geometrical eigenmodes’ (viz., B 1 and B 2), define the large-scale structure of the butterfly diagrams. Remaining SHF modes define the orderliness of the field distribution even within the ‘wings’ of the ‘butterflies’ down to scales l ≈ 29. These include the ‘geometrical eigenmodes’ B 3 and B 4, which are not present in simulated data sets in which the latitudes of the sunspot groups are randomly redistributed within the ‘wings’ of the ‘butterflies’. Superposition of B 1, B 2, B 3, and B 4 is necessary and sufficient to reproduce important observed properties of the latitude-time distribution of the real field, not only in the ‘sunspot zone’, but also in the middle (35°–75°) and the high (≳75°) latitudes, with appropriate relative orders of magnitude and phases. Thus, B 1, B 2, B 3, and B 4 seem to represent really existing global oscillations in the Sun's internal magnetic field. The geometrical form of B 1 may also be the form of the ‘forcing’ oscillation.