A Simple Model for the Solar Cycle

A simple nonlinear model which describes the 11-year solar cycle can be derived from the usual – dynamo theory in the form of a Van der Pol equation. Solar activity displays also small-scale inter-cycle persistent stochastic oscillations with a Hurst exponent of the order of H0.76±0.01. The results obtained from the Van der Pol oscillator superimposed on a fractional Brownian motion which describes the stochastic fluctuations are presented.     

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.     

The 27-Day and 22-Year Cycles in Solar and Geomagnetic Activity

We study the evolution of the longitudinal asymmetry in solar activity through the wave packet technique applied to the period domain of 25 – 31 days (centered at the 27-day solar rotation period) for the sunspot number and geomagnetic aa index. We observe the occurrence of alternating smaller and larger amplitudes of the 11-year cycle, resulting in a 22-year periodicity in the 27-day signal. The evolution of the 22-year cycle shows a change of regime around the year 1912 when the 22-year period disappears from the sunspot number series and appears in the aa index. Other changes, such as a change in the correlation between solar and geomagnetic activity, took place at the same time. Splitting the 27-day frequency domain of aa index shows an 11-year cycle for higher frequencies and a pure22-year cycle for lower frequencies, which we attribute to higher latitude coronal holes. This evidence is particularly clear after 1940, which is another benchmark in the evolution of the aa index. We discuss briefly the mechanisms that could account for the observed features of the 22-year cycle evolution.     

Short-Term Periodicities in Sunspot Activity and Flare Index Data during Solar Cycle 23

The short-term periodicities in sunspot numbers, sunspot areas, and flare index data are investigated in detail using the Date Compensated Discrete Fourier Transform (DCDFT) for the full disk of the Sun separately over the rising, the maximum, and the declining portions of solar cycle 23 (1996 – 2006). While sunspot numbers and areas show several significant periodicities in a wide range between 23.1 and 36.4 days, the flare index data do not exhibit any significant periodicity. The earlier conclusion of Pap, Tobiska, and Bouwer (1990, Solar Phys. 129, 165) and Kane (2003, J. Atmos. Solar-Terr. Phys. 65, 1169), that the 27-day periodicity is more pronounced in the declining portion of a solar cycle than in the rising and maximum ones, seems to be true for sunspot numbers and sunspot area data analyzed here during solar cycle 23.     

Search for Rapid Changes in the Visible-Light Corona during the 21?June 2001 Total Solar Eclipse

Some 8000 images obtained with the Solar Eclipse Coronal Imaging System (SECIS) fast-frame CCD camera instrument located at Lusaka, Zambia, during the total eclipse of 21 June 2001 have been analysed to search for short-period oscillations in intensity that could be a signature of solar coronal heating mechanisms by MHD wave dissipation. Images were taken in white-light and Fe xiv green-line (5303 ?) channels over 205 seconds (frame rate 39 s⁻¹), approximately the length of eclipse totality at this location, with a pixel size of four arcseconds square. The data are of considerably better quality than those that we obtained during the 11 August 1999 total eclipse (Rudawy et al.: Astron. Astrophys. 416, 1179, 2004), in that the images are much better exposed and enhancements in the drive system of the heliostat used gave a much improved image stability. Classical Fourier and wavelet techniques have been used to analyse the emission at 29 518 locations, of which 10 714 had emission at reasonably high levels, searching for periodic fluctuations with periods in the range 0.1 – 17 seconds (frequencies 0.06 – 10 Hz). While a number of possible periodicities were apparent in the wavelet analysis, none of the spatially and time-limited periodicities in the local brightness curves was found to be physically important. This implies that the pervasive Alfvén wave-like phenomena (Tomczyk et al.: Science 317, 1192, 2007) using polarimetric observations with the Coronal Multi-Channel Polarimeter (CoMP) instrument do not give rise to significant oscillatory intensity fluctuations.     

Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle

We present the results of a statistical study of the solar cycle based on the analysis of the superficial toroidal magnetic field component phase space. The magnetic field component used to create the embedded phase space was constructed from monthly sunspot number observations since 1750. The phase space was split into 32 sections (or time instants) and the average values of the orbits on this phase space were calculated (giving the most probable cycle). In this phase space it is shown that the magnetic field on the Sun’s surface evolves through a set of orbits that go around a mean orbit (i.e., the most probable magnetic cycle that we interpret as the equilibrium solution). It follows that the most probable cycle is well represented by a van der Pol oscillator limit curve (equilibrium solution), as can be derived from mean-field dynamo theory. This analysis also retrieves the empirical Gnevyshev – Ohl’s rule between the first and second parts of the solar magnetic cycle. The sunspot number evolution corresponding to the most probable cycle (in phase space) is presented.     

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.     

The Solar Ultraviolet Spectrum Estimated Using the Mg <Emphasis Type="SmallCaps">ii</Emphasis> Index and Ca <Emphasis Type="SmallCaps">ii</Emphasis> K Disk Activity

As part of a program to estimate the solar spectrum back to the early twentieth century, we have generated fits to UV spectral irradiance measurements from 1 – 410 nm. The longer wavelength spectra (150 – 410 nm) were fit as a function of two solar activity proxies, the Mg ii core-to-wing ratio, or Mg ii index, and the total Ca ii K disk activity derived from ground based observations. Irradiance spectra at shorter wavelengths (1 – 150 nm) where used to generate fits to the Mg ii core-to-wing ratio alone. Two sets of spectra were used in these fitting procedures. The fits at longer wavelengths (150 to 410 nm) were derived from the high-resolution spectra taken by the Solar Ultraviolet Spectral Irradiance Monitor (SUSIM) on the Upper Atmospheric Research Satellite (UARS). Spectra measured by the Solar EUV Experiment (SEE) instrument on the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite were used for the fits at wavelengths from 1 to 150 nm. To generate fits between solar irradiance and solar proxies, this study uses the above irradiance data, the NOAA composite Mg ii index, and daily Ca ii K disk activity determined from images measured by Big Bear Solar Observatory (BBSO). In addition to the fitting coefficients between irradiance and solar proxies, other results from this study include an estimated relationship between the fraction of the disk with enhanced Ca ii K activity and the Mg ii index, an upper bound of the average solar UV spectral irradiance during periods where the solar disk contains only regions of the quiet Sun, as was believed to be present during the Maunder Minimum, as well as results indicating that slightly more than 60% of the total solar irradiance (TSI) variability occurs between 150 and 400 nm.     

Magnetic Flux Imbalance of the Solar and Heliospheric Magnetic Fields

A comparative analysis of solar and heliospheric magnetic fields in terms of their cumulative sums reveals cyclic and long-term changes that appear as a magnetic flux imbalance and alternations of dominant magnetic polarities. The global magnetic flux imbalance of the Sun manifests itself in the solar mean magnetic field (SMMF) signal. The north – south asymmetry of solar activity and the quadrupole mode of the solar magnetic field contribute the most to the observed magnetic flux imbalance. The polarity asymmetry exhibits the Hale magnetic cycle in both the radial and azimuthal components of the interplanetary magnetic field (IMF). Analysis of the cumulative sums of the IMF components clearly reveals cyclic changes in the IMF geometry. The accumulated deviations in the IMF spiral angle from its nominal value also demonstrate long-term changes resulting from a slow increase of the solar wind speed over 1965 – 2006. A predominance of the positive IMF B _z with a significant linear trend in its cumulative signal is interpreted as a manifestation of the relic magnetic field of the Sun. Long-term changes in the IMF B _z are revealed. They demonstrate decadal changes owing to the 11/22-year solar cycle. Long-duration time intervals with a dominant negative B _z component were found in temporal patterns of the cumulative sum of the IMF B _z .     

Fluctuations of the Solar Dynamo Observed in the Solar Wind and Interplanetary Magnetic Field at 1 AU and in the Outer Heliosphere

Recent helioseismic observations have found strong fluctuations at a period of about 1.3 years in the rotation speed around the tachocline in the deep solar convection layer. Similar mid-term quasi-periodicities (MTQP; periods between 1–2 years) are known to occur in various solar atmospheric and heliospheric parameters for centuries. Since the deep convection layer is the expected location of the solar magnetic dynamo, its fluctuations could modulate magnetic flux generation and cause related MTQP fluctuations at the solar surface and beyond. Accordingly, it is likely that the heliospheric MTQP periodicities reflect similar changes in solar dynamo activity. Here we study the occurrence of the MTQP periodicities in the near and distant heliosphere in the solar wind speed and interplanetary magnetic field observed by several satellites at 1 AU and by four interplanetary probes (Pioneer 10 and 11 and Voyager 1 and 2) in the outer heliosphere. The overall structure of MTQP fluctuations in the different locations of the heliosphere is very consistent, verifying the solar (not heliospheric) origin of these periodicities. We find that the mid-term periodicities were particularly strong during solar cycle 22 and were observed at two different periods of 1.3 and 1.7 years simultaneously. These periodicities were latitudinally organized so that the 1.3-year periodicity was found in solar wind speed at low latitudes and the 1.7-year periodicity in IMF intensity at mid-latitudes. While all heliospheric results on the 1.3-year periodicity are in a good agreement with helioseismic observations, the 1.7-year periodicity has so far not been detected in helioseismic observations. This may be due to temporal changes or due to the helioseismic method where hemispherically antisymmetric fluctuations would so far have remained hidden. In fact, there is evidence that MTQP fluctuations may occur antisymmetrically in the northern and southern solar hemisphere. Moreover, we note that the MTQP pattern was quite different during solar cycles 21 and 22, implying fundamental differences in solar dynamo action between the two halves of the magnetic cycle.     

Spectro-Polarimetric Observations of Solar Magnetic Fields and the SOHO/MDI Calibration Issue

Comparisons of solar magnetic-field measurements made in different spectral lines are very important, especially in those lines in which observations have a long history or (and) specific diagnostic significance. The spectral lines Fe i 523.3 nm and Fe i 525.0 nm belong to this class. Therefore, this study is devoted to a comprehensive analysis using new high-precision Stokes-meter full-disk observations. The disk-averaged magnetic-field strength ratio R=B(523.3)/B(525.0) equals 1.97±0.02. The center-to-limb variation (CLV) is R=1.74−2.43μ+3.43μ ², where μ is the cosine of the center-to-limb angle. For the disk center, we find R=2.74, and for near-limb areas with μ=0.3, R equals 1.32. There is only a small dependence of R on the spatial resolution. Our results are rather close to those published three decades ago, but differ significantly from recent magnetographic observations. An application of our results to the important SOHO/MDI magnetic data calibration issue is discussed. We conclude that the revision of the SOHO/MDI data, based only on the comparison of magnetic-field measurements in the line pair Fe i 523.3 nm and Fe i 525.0 nm (increasing by a factor of 1.7 or 1.6 on average according to recent publications) is not obvious and new investigations are urgently needed.     

Short-Term Periodicities in Sunspot Activities During the Descending Phase of Solar Cycle 23

In the present study, the short-term periodicities in the daily data of the sunspot numbers and areas are investigated separately for the full disk, northern, and southern hemispheres during Solar Cycle 23 for a time interval from 1 January 2003 to 30 November 2007 corresponding to the descending and minimum phase of the cycle. The wavelet power spectrum technique exhibited a number of quasi-periodic oscillations in all the datasets. In the high frequency range, we find a prominent period of 22 – 35 days in both sunspot indicators. Other quasi-periods in the range of 40 – 60, 70 – 90, 110 – 130, 140 – 160, and 220 – 240 days are detected in the sunspot number time series in different hemispheres at different time intervals. In the sunspot area data, quasi-periods in the range of 50 – 80, 90 – 110, 115 – 130, 140 – 155, 160 – 190, and about 230 days were noted in different hemispheres within the time period of analysis. The present investigation shows that the well-known “Rieger periodicity” of 150 – 160 days reappears during the descending phase of Solar Cycle 23, but this is prominent mainly in the southern part of the Sun. Possible explanations of these observed periodicities are delivered on the basis of earlier results detected in photospheric magnetic field time series (Knaack, Stenflo, and Berdyugina in Astron. Astrophys. 438, 1067, 2005) and solar r-mode oscillations.     

Lifetimes of High-Degree <Emphasis Type="Italic">p</Emphasis> Modes in the Quiet and Active Sun

We study variations of the lifetimes of high-ℓ solar p modes in the quiet and active Sun with the solar activity cycle. The lifetimes in the degree range ℓ=300 – 600 and ν=2.5 – 4.5 mHz were computed from SOHO/MDI data in an area including active regions and quiet Sun using the time – distance technique. We applied our analysis to the data in four different phases of solar activity: 1996 (at minimum), 1998 (rising phase), 2000 (at maximum), and 2003 (declining phase). The results from the area with active regions show that the lifetime decreases as activity increases. The maximal lifetime variations are between solar minimum in 1996 and maximum in 2000; the relative variation averaged over all ℓ values and frequencies is a decrease of about 13%. The lifetime reductions relative to 1996 are about 7% in 1998 and about 10% in 2003. The lifetime computed in the quiet region still decreases with solar activity, although the decrease is smaller. On average, relative to 1996, the lifetime decrease is about 4% in 1998, 10% in 2000, and 8% in 2003. Thus, measured lifetime increases when regions of high magnetic activity are avoided. Moreover, the lifetime computed in quiet regions also shows variations with the activity cycle.     

The Imaging Vector Magnetograph at Haleakalā IV: Stokes Polarization Spectra in the Sodium D<Subscript>1</Subscript> 589.6 nm Spectral Line

The Imaging Vector Magnetograph (IVM) at the Mees Solar Observatory, Haleakalā, Maui, Hawai’i, obtained many years of vector magnetic-field data in the photospheric Fe i 630.25 nm line. In the latter period of its operation, the IVM was modified to allow routine observations in the chromospheric Na i D₁ line, as well as the Fe i line. We describe the sodium observational data in detail, including the data-reduction steps that differ from those employed for the Fe i 630.25 nm line, to obtain calibrated Stokes polarization spectra. We have performed a systematic comparison between the observational data and synthetic NLTE Na i D₁ Stokes spectra derived for a variety of solar-appropriate atmospheric and magnetic configurations. While the Na i D₁ Stokes polarization signals from the solar atmosphere are expected to be weak, they should generally be within the IVM capability. A comparison between synthetic spectra and observational data indicates that this is indeed the case.     

Predicting the Amplitude of a Solar Cycle Using the North – South Asymmetry in the Previous Cycle: II. An Improved Prediction for Solar Cycle 24

Recently, using Greenwich and Solar Optical Observing Network sunspot group data during the period 1874 – 2006, Javaraiah (Mon. Not. Roy. Astron. Soc. 377, L34, 2007: Paper I), has found that: (1) the sum of the areas of the sunspot groups in 0° – 10° latitude interval of the Sun’s northern hemisphere and in the time-interval of −1.35 year to +2.15 year from the time of the preceding minimum of a solar cycle n correlates well (corr. coeff. r=0.947) with the amplitude (maximum of the smoothed monthly sunspot number) of the next cycle n+1. (2) The sum of the areas of the spot groups in 0° – 10° latitude interval of the southern hemisphere and in the time-interval of 1.0 year to 1.75 year just after the time of the maximum of the cycle n correlates very well (r=0.966) with the amplitude of cycle n+1. Using these relations, (1) and (2), the values 112±13 and 74±10, respectively, were predicted in Paper I for the amplitude of the upcoming cycle 24. Here we found that the north – south asymmetries in the aforementioned area sums have a strong ≈44-year periodicity and from this we can infer that the upcoming cycle 24 will be weaker than cycle 23. In case of (1), the north – south asymmetry in the area sum of a cycle n also has a relationship, say (3), with the amplitude of cycle n+1, which is similar to (1) but more statistically significant (r=0.968) like (2). By using (3) it is possible to predict the amplitude of a cycle with a better accuracy by about 13 years in advance, and we get 103±10 for the amplitude of the upcoming cycle 24. However, we found a similar but a more statistically significant (r=0.983) relationship, say (4), by using the sum of the area sum used in (2) and the north – south difference used in (3). By using (4) it is possible to predict the amplitude of a cycle by about 9 years in advance with a high accuracy and we get 87±7 for the amplitude of cycle 24, which is about 28% less than the amplitude of cycle 23. Our results also indicate that cycle 25 will be stronger than cycle 24. The variations in the mean meridional motions of the spot groups during odd and even numbered cycles suggest that the solar meridional flows may transport magnetic flux across the solar equator and potentially responsible for all the above relationships. The author did a major part of this work at the Department of Physics and Astronomy, UCLA, 430 Portola Plaza, Los Angeles, CA 90095-1547, USA.     

Coupled <Emphasis Type="Italic">G</Emphasis>-Mode Intersections and Solar-Cycle Variability

Wolff (Astrophys. J. 193, 721, 1974) introduced the concept of g-mode coupling within the solar interior. Subsequently, Wolff developed a more quantitative model invoking a reciprocal interaction between coupled g modes and burning in the solar core. Coupling is proposed to occur for constant values of the spherical harmonic degree [ℓ] creating rigidly rotating structures denoted as sets(ℓ). Power would be concentrated near the core and the top of radiative zone [RZ] in narrow intervals of longitude on opposite sides of the Sun. Sets(ℓ) would migrate retrograde in the RZ as function of ℓ and their intersections would deposit extra energy at the top of the RZ. It is proposed that this enhances sunspot eruptions at particular longitudes and at regular time intervals. Juckett and Wolff (Solar Phys. 252, 247, 2008) detected this enhancement by viewing selected spherical harmonics of sunspot patterns within stackplots twisted into the relative rotational frames of various sets(ℓ). In subsequent work, the timings of the set(ℓ) intersections were compared to the sub-decadal variability of the sunspot cycle. Seventeen sub-decadal intersection frequencies (0.63 – 7.0 year) were synchronous with 17 frequencies in the sunspot time-series with a mean correlation of 0.96. Six additional non-11-year frequencies (periods of 8.0 to 28.7 year) are now shown to be nearly synchronous between sunspot variability and the model. Two additional intersections have the same frequency as the solar cycle itself and peak during the rising phase of the solar cycle. This may be partly responsible for cycle asymmetry. These results are evidence that some of the solar-cycle variability may be attributable to deterministic components that are intermixed with a broad-spectrum stochastic and long-term chaotic background.     

Weak Emission Lines in the Wings of Solar H and K

The rare-earth ions cerium ii, lanthanum ii, dysprosium ii, and additionally zirconium ii and iron ii, are seen as weak emission features in the wings of the solar Ca ii H and K lines. The strength of these emission lines increases on the disk toward the limb. We provide recent high-resolution observations at disk center and at the limb. The identity of the weak lines is re-worked. We point out the unique role of eclipse spectra in distinguishing between the photospheric and chromospheric origins of emission lines. It is then demonstrated from our full disk (Sun-as-a-Star) and center disk archives, 1974 – 2010, that no activity cycle related signal is evident (save for the H and K lines themselves).     

Latitude Dependence of the “Gnevyshev” Peaks and Gaps

The occurrence of double peaks near the maximum of sunspot activity was first emphasized by Gnevyshev (Solar Phys. 1, 107, 1967) for the peak years of solar cycle 19 (1954 – 1964). In the present analysis, it is shown that double peaks in sunspot numbers were clearly visible in solar latitudes 10 – 30° N but almost absent in the southern latitudes, where some single peaks were observed out of phase by several months from any of the peaks in the northern latitudes. The spacing between the double peaks increased from higher to lower northern latitudes, hinting at latitudinal migration. In the next cycle 20 (1965 – 1976), which was of about half the strength of cycle 19, no clear-cut double peaks were seen, and the prominent peak in the early part of 1967 in the northern latitudes was seen a few months later in the southern latitudes. A direct relationship of Gnevyshev peaks with changes in the solar polar magnetic fields seems to be dubious. The commencements do not match.     

Solar Cycle Predictions based on Solar Activity at Different Solar Latitudes

Many methods of predictions of sunspot maximum number use data before or at the preceding sunspot minimum to correlate with the following sunspot maximum of the same cycle, which occurs a few years later. Kane and Trivedi (Solar Phys. 68, 135, 1980) found that correlations of R _z(max) (the maximum in the 12-month running means of sunspot number R _z) with R _z(min) (the minimum in the 12-month running means of sunspot number R _z) in the solar latitude belt 20° – 40°, particularly in the southern hemisphere, exceeded 0.6 and was still higher (0.86) for the narrower belt > 30° S. Recently, Javaraiah (Mon. Not. Roy. Astron. Soc. 377, L34, 2007) studied the relationship of sunspot areas at different solar latitudes and reported correlations 0.95 – 0.97 between minima and maxima of sunspot areas at low latitudes and sunspot maxima of the next cycle, and predictions could be made with an antecedence of more than 11 years. For the present study, we selected another parameter, namely, SGN, the sunspot group number (irrespective of their areas) and found that SGN(min) during a sunspot minimum year at latitudes > 30° S had a correlation +0.78±0.11 with the sunspot number R _z(max) of the same cycle. Also, the SGN during a sunspot minimum year in the latitude belt (10° – 30° N) had a correlation +0.87±0.07 with the sunspot number R _z(max) of the next cycle. We obtain an appropriate regression equation, from which our prediction for the coming cycle 24 is R _z(max )=129.7±16.3.     

A Phenomenological Study of the Cosmic Ray Variations over the Past 9400 Years, and Their Implications Regarding Solar Activity and the Solar Dynamo

Two 9400-year long ¹⁰Be data records from the Arctic and Antarctic and a ¹⁴C record of equal length were used to investigate the periodicities in the cosmic radiation incident on Earth throughout the past 9400 years. Fifteen significant periodicities between 40 and 2320 years are observed in the ¹⁰Be and ¹⁴C records, there being close agreement between the periodicities in each record. We found that the periodic variations in the galactic cosmic radiation are the primary cause for periods <?250 years, with minor contributions of terrestrial origin possible >?250 years. The spectral line for the Gleissberg (87-year) periodicity is narrow, indicating a stability of ≈?0.5 %. The 9400-year record contains 26 Grand Minima (GM) similar to the Maunder Minimum, most of which occurred as sequences of 2?–?7 GM with intervals of 800?–?1200 years in between, in which there were no GM. The intervals between the GM sequences are characterised by high values of the modulation function. Periodicities <?150 years are observed in both the GM intervals and the intervals in between. The longer-period variations such as the de Vries (208-year) cycle have high amplitudes during the GM sequences and are undetectable in between. There are three harmonically related pairs of periodicities (65 and 130 years), (75 and 150 years), and (104 and 208 years). The long periodicities at 350, 510, and 708 years closely approximate 4, 6, and 8 times the Gleissberg period (87 years). The well-established properties of cosmic-ray modulation theory and the known dependence of the heliospheric magnetic field on the solar magnetic fields lead us to speculate that the periodicities evident in the paleo-cosmic-ray record are also present in the solar magnetic fields and in the solar dynamo. The stable, narrow natures of the Gleissberg and other periodicities suggest that there is a strong “frequency control” in the solar dynamo, in strong contrast to the variable nature (8?–?15 years) of the Schwabe (11-year) solar cycle.