Citation:Feynman, J., and A. Ruzmaikin (2014),The Centennial Gleissberg Cycle andits association with extended minima,J. Geophys. Res. SpacePhysics, 119, 6027–6041,doi:10.1002/2013JA019478.
Received 26 SEP 2013
Accepted 29 JUL 2014
Accepted article online 4 AUG 2014
Published online 25 AUG 2014
The Centennial Gleissberg Cycle and its associationwith extended minimaJ. Feynman1 and A. Ruzmaikin1
1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA
Abstract The recent extended minimum of solar and geomagnetic variability (XSM) mirrors the XSMsin the nineteenth and twentieth centuries: 1810–1830 and 1900–1910. Such extended minima also wereevident in aurorae reported from 450 A.D. to 1450 A.D. This paper argues that these minima are consistentwith minima of the Centennial Gleissberg Cycles (CGCs), a 90–100 year variation observed on the Sun, in thesolar wind, at the Earth, and throughout the heliosphere. The occurrence of the recent XSM is consistentwith the existence of the CGC as a quasiperiodic variation of the solar dynamo. Evidence of CGCs is providedby the multicentury sunspot record, by the almost 150 year record of indexes of geomagnetic activity(1868 to present), by 1000 years of observations of aurorae (from 450 to 1450 A.D.) and millennial recordsof radionuclides in ice cores. The aa index of geomagnetic activity carries information about the twocomponents of the solar magnetic field (toroidal and poloidal), one driven by flares and coronal massejections (related to the toroidal field) and the other driven by corotating interaction regions in the solarwind (related to the poloidal field). These two components systematically vary in their intensity and relativephase giving us information about centennial changes of the sources of solar dynamo during the recentCGC over the last century. The dipole and quadrupole modes of the solar magnetic field changed in relativeamplitude and phase; the quadrupole mode became more important as the XSM was approached. Someimplications for the solar dynamo theory are discussed.
1. Introduction
Observations of sunspots, geomagnetic activity, aurorae, and radionuclides have been used to deduce thepast behavior of the Sun. For example, Eddy [1976] reviewed the seventeenth to twentieth century litera-ture and showed that almost no sunspot or auroral reports were recorded between 1645 A.D. and 1715 A.D.,a period he called the Maunder Minimum. It took many years before the scientific community acceptedhis results. Now the Maunder Minimum is known to be one of many Grand Minima and Maxima that haveoccurred in the past [Usoskin, 2013]. Although less well known, but well established is the 90–100 yearperiod variation in the “11 year” cycle amplitude throughout history [Gleissberg, 1965; Siscoe, 1980; Keimatsu,1976]. It is persistent in the about 12,000 year accurately calibrated record of atmospheric 14C [Peristykh andDamon, 2003] and in the 9000 year records of 10Be in Arctic and Antarctic polar ice and in 14C from a globalcompilation of tree rings and marine sediments [McCracken et al., 2013; Beer et al., 2012]. These papershave also reported longer cycles, such as the 208 year de Vries cycle and the 2200 year Hallstatt cycle, notdiscussed in this paper.
Numerous investigations of the solar output have led to the suggestion that there are variations with 50and 90–100 years in the amplitude of the solar cycle collectively referred to as Gleissberg Cycles [Gleissberg,1965; Ogurtsov et al., 2002; Javaraiah et al., 2005; Demetrescu and Dobrica, 2008]. The 90–100 year cycle hasbeen studied extensively [Feynman and Crooker, 1978; Feynman and Fougere, 1984; Lockwood et al., 1999;Pulkkinen et al., 2001; Mursula and Martini, 2006] and called the Centennial Gleissberg Cycle (CGC) [Feynmanand Ruzmaikin, 2011] to distinguish it from the other modulations of the 11 year cycle amplitude such asthe double Hale cycle [Javaraiah et al., 2005]. The minima of the CGC should be distinguished from GrandMinima [Usoskin et al., 2007]. There were reports of the CGC presence in the record of the nitrate depositionin an ice core from Greenland in 1561–1950 interpreted as caused by solar proton events [McCracken et al.,2001]. However, the follow-up investigations showed that polar ice sheet nitrates in Greenland are moreclosely associated with forest fires in USA/Canada than with solar proton events [Wolff et al., 2012; Schrijveret al., 2012].
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The sunspot record during the last 300 years shows changes in the amplitude and period of the sunspotcycle. In particular, there were three times when the sunspot cycle had smaller maxima and deeper andlonger minima than usual: when exiting the Maunder Minimum early in the eighteenth century, early in thenineteenth century, and the twentieth centuries. This led Silverman [1992] to speculate that another suchevent could take place at the beginning of the 21st century. Indeed, the Sun during the transition betweenthe solar cycles 23 and 24 exhibited an extended, deep minimum that has been widely commented upon[Gibson et al., 2011; Lee et al., 2009]. During that minimum in 2008 there were 250 days without sunspots.This was the second quietest year in over 100 years [Feynman and Ruzmaikin, 2011, 2012]. Only the twen-tieth century deep minimum was quieter with over 300 spotless days in 1913, and 250 days in 1912. Thetime and amplitude of the 24th sunspot cycle maximum itself cannot yet be determined because, as of thiswriting (23 July 2014), it is not yet established that the cycle maximum has passed.
In this paper we present evidence that the recent extended solar minimum (XSM) is consistent withthe minimum of the Centennial Gleissberg Cycle. The existence of the CGC is now established by theanalyses of millennium long ice core records, by thousand year long auroral records, and now with morecomplete contemporary records of solar and solar-terrestrial observations that cover more than an entirecentennial cycle.
Section 2 describes the observed changes in the 11 year sunspot cycles as the CGC proceeds. Section 3reviews the changes in the solar wind over the last four solar cycles related to this CGC. Section 4 presentsevidence of the CGC recorded in geomagnetic activity. Section 5 summarizes the analysis of the observa-tional data. Section 6 discusses processes on the Sun that could explain the origin of the CGC.
2. CGC on the Sun
Evidence of the CGC in the long-term record of sunspots is shown in Figure 1. Figure 1a shows the annuallyaveraged sunspot number (SSN) from the time the Sun recovered from the Maunder Minimum period ofinactivity [Eddy, 1976] until the present. There is a centennial variation in the amplitude of the sunspot cyclethroughout this history. This amplitude modulation is a major aspect of the CGC. Several 11 year Schwabecycle minima are denoted by arrows corresponding to years with the annual sunspot numbers 3 or less.These very low sunspot minima appear in groups over one or more decades. We can define the CGC minimalphase by these decades.
Figure 1b shows a time-dependent (wavelet) spectrum of the data shown in Figure 1a. The 11 year sunspotcycle and the 100 year CGC on this graph are marked with solid lines. The panel also shows a quasiperiodiccycle beginning at about 1725 with a periodicity of about 50–60 years drifting to lower frequencies untilwhen at about 1850 it merges with the CGC. This variation does not persist and will not be discussed furtherin this paper. The integral spectrum shown in Figure 1c is obtained by averaging the wavelet spectrum. Thecentennial periodicity is only marginally significant (at 1𝜎 level) because of limited length of the time series.It is more significant when the long time series such as 10Be and 14C records are used [McCracken et al., 2013;Beer et al., 2012].
Figure 1d details the spectral change of the solar (SSN) cycle. Note that the periods of the sunspot cycles(cycle lengths) appear to be longer during the XSMs indicating (because of a low level of statistical signifi-cance) the frequency modulation of the solar cycle by the CGC. For example, during the first few solar cyclesof the eighteenth century the SSN period was about 11 years. It then decreased to about 9 years. This wasfollowed by the sudden increase to over 12 years during the first two cycles of the nineteenth century, atime known as the Dalton minimum. (We note that there is some ambiguity of the solar cycle definitionaround the Dalton minimum, including the “lost cycle” [Usoskin et al., 2009].) Like the recent XSM, the min-ima between these two cycles were extended as can be seen in Figure 1a. Although not as dramatic, thereare also cycles with relatively long periods and XSMs between 1890 and 1915. Addressing a possible physicalconnection between the cycle length and its amplitude, we mention that Dikpati et al. [2010] found a stronganticorrelation between the length and depth of solar cycle minima and suggested that this is becauseoppositely directed toroidal bands in close proximity on the two sides of the equator have more time in thecase of a longer minimum to annihilate each other, leading to fewer or no eruptions of low-altitude spots.
Figure 1e shows the wavelet mode filtered in the 80–100 year scale range. This mode corresponds to theCGC showing four deep solar minima. Using this mode, we estimate the CGC period during the last 400 years
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Figure 1. (a) The annual sunspot number record in 1700–2012 assembled by the Belgium Solar Influences Data AnalysisCenter (http://www.sidc.be). Each arrow and asterisk denotes times when the annual sunspot number was very low (lessthan 3) indicating the CGC minima, see the text. (b) Wavelet spectrum of SSN. The edge effects, which may distort thewavelet spectrum, have no influence above the dashed line [Torrence and Compo, 1998]. Solid lines mark the 11 yearand 100 year periods. (c) The integral spectrum obtained by averaging over the time axis. The dashed line shows thesignificance of this spectrum at the 1𝜎 level. (d) The detailed wavelet spectrum in the 5.2 –17.4 period region. (e) Thetime series of the 80–110 year band.
as about 100 years, slightly longer than the mean period of about 90 years estimated from the analysis ofEuropean and Chinese midlatitude aurora observed between the years 450 and 1450 [Siscoe, 1980; Feynmanand Fougere, 1984].
Since solar activity is driven by motions and generated by the magnetic fields in the solar convection zone,it is desirable to look into the long-term records of the solar motions.
Pulkkinen and Tuominen [1998] found a centennial change in the surface differential rotation using theobservations of sunspot rotation across solar latitudes in 1853–1996. The largest change in the differen-tial rotation occurred around the thirteenth solar cycle (1890–1894), which is ahead of but close to the CGCtwentieth century minimum. The solar differential rotation measurements were continued to 1998–2006 byWöhl et al. [2010] (see also references in that paper) showing some increase in the rate of rotation and con-firming the north-south asymmetry of the rotation indicated by Pulkkinen and Tuominen [1998]. Javaraiah etal. [2005], using the sunspot group data, have shown that the latitudinal gradient of the solar rotation showsa significant modulation of about 79 years, which is consistent with what is expected for the existence ofthe Centennial Gleissberg Cycle. Although these papers deal with the surface rotation, the helioseismologyshowed that the surface rotation is tightly related to the rotation at the tachocline (at the heart of dynamo)[Schou, 1998]. Taking into account the angular momentum transport forming the radial-latitudinal profile
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of the solar rotation [Pipin, 1999], it is reasonable to expect that the changes on the solar surface reflect thechanges at the tachocline.
Observations also show a north-south asymmetry of the sunspot distribution. For example, the magneticequator defined by the average of the signed value of the latitude of each sunspot (positive for north, nega-tive for south) oscillates with a period of about 90 years and a mean amplitude of 1.3◦ [Pulkkinen et al., 1999].It shows a CGC variation in such a way that the sunspot belts are most southerly during the early twentiethcentury CGC, switch to more northerly with a maximum around 1950, and then begin to move to the southagain after 1950. The variation of the sunspot area for 130 years shows a long-term north-south phasesynchronization [Zolotova et al., 2009]. Since the sunspots are associated with the toroidal magnetic field,these observations provide indirect information about the change in the north-south asymmetry in thefield distribution. The link between north-south asymmetry, cycle length, and Gleissberg Cycles in dynamobehavior has been discussed by Tobias [2002].
There are no direct long-term records of solar magnetic fields. While there were some episodic measure-ments of the sunspot magnetic fields, the systematic measurements of the solar magnetic field started in1976 at Wilcox Solar Observatory. However, even this relatively short record shows that the polar magneticfield at the solar cycle minima have been systematically decreasing for the last 25 years from about 1.3 G in1986 at the cycle 21/22 minimum to about 0.5 G during the recent XSM in 2009 [Feynman and Ruzmaikin,2011]. Figure 2 displays the time behavior of two major modes of the solar photospheric magnetic field, thedipole and quadrupole modes. Note that although both modes are decreasing, the decrease in the dipoleis much larger than that of the quadrupole, and the phase between the modes is changing. According toclassical dynamo theories [cf. Parker, 1979], the dipole and quadrupole maximize at the solar minima. Wesee that indeed the dipole was near zero at the end of 2012 and at the beginning of the solar maximum in1979, but the quadrupole component of the magnetic field shows a rise toward the current maximum andthe 1979 maximum.
The Fourier analysis of these modes does not allow us to distinguish between their periods due the shorttime series. An application of a more sensitive nonlinear empirical mode decomposition technique [Wuand Huang, 2009] reveals that they oscillate with slightly different distributions of periods as shown inFigure 2, but we must wait for longer records to make statistically significant conclusions. The importance ofknowing the difference between the periods of oscillation of these two magnetic modes will be discussedin section 6.
3. CGC in the Solar Wind.
The solar wind arises from the Sun, travels through space, and interacts with the magnetic field of the Earth,producing geomagnetic activity and aurora. The interplanetary parameters of the solar wind have beenobserved in space in the vicinity of the Earth since 1964, and the data are available in the OMNI web site(http://omniweb.gsfc.nasa.gov/). Figure 3 displays the 27 day averages of the dynamic pressure and themagnitude of interplanetary magnetic field, both of which show systematic changes during the 49 yearsof data. Figure 3a shows the sunspot cycle variation for timing comparison. The recent XSM and the weaksolar cycle following it appear on the right of the figure. A comparison of the dynamic pressure during thesolar minima of 1975 and 1985 and 2008 reveals a dramatic change from about 3 nPa in 1975 to near 1 nPain 2008. The interplanetary field intensity during this period shown in Figure 3c dropped by a factor of 3.The changes were accomplished gradually over four solar cycles showing again that the XSM was a part of asystematic change that took place over almost half a century.
To better demonstrate the difference between the solar wind before and after the onset of the XSM we eval-uate the joint probability distribution of these two solar wind parameters. Figure 4a shows contours of thejoint distribution of the 27 day averages of the magnetic field intensity and dynamic pressure. Figure 4bshows the joint distribution since the current XSM began in 2006 and clearly indicates that the solar windthen was distinct from the wind observed in situ for the previous solar minima, such as the shown distri-bution at the minimum around 1976. An equivalent behavior had been reported [Feynman and Crooker,1978] for the twentieth century XSM in 1901. It was known that the annual averaged aa index (see section 4)exhibited correlation of 0.9 with the square of annual solar wind velocity times the annual southward com-ponent of the magnetic field [Crooker et al., 1977]. Feynman [1982] showed that the aa index in 1901 couldnot be determined from a sample of a solar wind measured during the space age in solar cycle 20. Thus,
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Figure 2. The dipole (solid curves) and quadrupole (dash curves) of the solar magnetic field obtained from the l = 1and l = 2 harmonics from the Wilcox Solar Observatory. (a) Data; (b) data smoothed with 27 day running average; (c)data filtered with empirical mode decomposition (EMD) code; the units in Figures 2a–2c are in Gauss. (d) Distributionfunctions of instantaneous frequencies of the dipole and quadrupole as functions of periods of their oscillations. Thedistributions differ in width and positions of their maxima.
the two XSMs of the 20th and 21st centuries share the attribute that the solar wind during both XSMs wasdifferent from that observed in situ during more typical solar cycles.
Recently, Lockwood and Owens [2014] and Lockwood et al. [2014] have reconstructed near-Earth interplan-etary magnetic field B and solar wind speed V for 1868 to the present using four different combinations ofgeomagnetic activity indices (including IHV (interhour variability index) and corrected aa index) which givealmost identical results. They show that the differences between IHV and the corrected aa are not due touncertainties in either index but because they respond slightly differently to B and V. Lockwood et al. [2014]also put the full 2𝜎 errors on these reconstructions using a Monte Carlo fitting technique and found thatthey are remarkably small.
4. Geomagnetic Records of Centennial Solar Variability.
The solar wind drives geomagnetic activity and geomagnetic storms are a reflection of disturbance pat-terns in the wind caused by the Sun. Two main types of storms are routinely distinguished [Newton, 1948]:storms that are associated with flares and coronal mass ejections (CMEs) [Carrington, 1860; Hirshberg and
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Figure 3. The 27 day averages of (a) sunspot numbers, (b) solar wind flow pressure, and (c) interplanetary magnetic fieldin 1964–2013. The data are from http://omniweb.gsfc.nasa.gov.
Colburn, 1969], which begin suddenly and do not repeat after 27 days, and storms related to coronal holes,which begin gradually and have a strong tendency to repeat each 27 days [Neupert and Pizzo, 1974]. Notethat the term “coronal mass ejection” (CME), which was introduced by Hundhausen et al. [1984], was notknown in those historical times. This classical view implies that there is a component of geomagnetic activ-ity related to the sunspot number and another component related to the coronal holes [Feynman and Gu,1986]. The solar wind is also associated with auroras, and auroras seen at midlatitudes are driven by coronalmass ejections.
The long-term record of geomagnetic activity is well characterized by the aa index [Mayaud, 1980]; a mea-sure of the range of geomagnetic variations seen at midlatitudes on the Earth. It is imperative to comparethe behavior of the aa index and the sunspot number from 1868 to 2013 (Figures 1a and 5a). The sunspotnumber returns to a value near zero at each solar minimum. In contrast, the values of aa at solar minima,with the exception of the 1964 SSN minimum, increase between 1900 until the middle of the century, thendecrease from the 1970 to the recent XSM.
The aa index reflects the character of the solar wind observed near the Earth, it depends on solar windspeed (squared), and the intensity and direction of the magnetic field. It is highly related to the often usedAp and Kp indices, see Mayaud [1980] for the definitions of these indices and the relations among them. Fora current review of all indices see Lockwood [2013]. The aa data set exists from 1868 to the present (Figure 5).Svalgaard and Cliver [2007] have introduced a new index (IHV), which displays a slightly different responseto the parameters of the solar wind.
The response of the magnetosphere to the solar wind depends only on the parameters of the solar windarriving at 1 AU. The relationship between the parameters in the solar wind and the aa index has been stud-ied since the first observations of the solar wind were reported. Perhaps the most extensive and reliablestudy of the expression that couples the solar wind to the geomagnetic variations available today is that
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Figure 4. The joint probability density function (pdf ) of the interplanetary magnetic field and dynamical pressure con-structed from the data shown in Figure 3. (a) The pdf during the solar minimum in 1974–1977 preceding the currentGleissberg minimum. (b) The pdf in the time of CGC minimum 2006–2013. The pdfs clearly differ in magnitude andposition. All pdfs are normalized to 1 and then multiplied by a factor of 103 for better display.
of Stamper et al. [1999]. From a study of the aa index Lockwood et al. [1999] estimated that the Sun’s coro-nal magnetic field had doubled during the first half of the twentieth century. A controversy developed inwhich the validity of the Mayaud’s aa index was reexamined by Svalgaard and Cliver [2007], see, however,[Lockwood et al., 2006]. The low values for the geomagnetic activity during the early twentieth century werechallenged as unreasonable and/or incorrect because no comparable annual aa values had been observedup to the time of these publications. That argument is muted by the observations of the aa during therecent XSM (Figure 5). Figure 5b shows a comparison between aa from 1897 to 1905 with the aa index from2005 to 2012, i.e, for the two XSMs. The IHV index is also shown for comparison. Not only are the aa valuesequal at the two XSMs but the similarity extends over 7 years. This implies that the Sun’s coronal magneticfield returned to the values it had at the beginning of the twentieth century so that we have now for the firsttime observed a full CGC of aa.
The beginning of the nineteenth century was a period of very weak solar cycles called the Dalton minimumoften erroneously referred to as a short return of the Maunder Minimum. The coldest part of the Daltonminimum is easily understood as the almost simultaneous effect of a normal XSM and the cooling effect ofthe material ejected by the volcanic eruption of Mount Tambora on the island in the Indian Ocean in April1815. The sunspot number for the 1804 and 1816 maxima were low (47 and 46, correspondingly), and thecycle length was extended for both the Dalton CGC minimum and the next CGC minimum, strongly in thenineteenth century and comparatively less so in the twentieth century.
For the Dalton XSM we have SSN data but no aa index data because the geomagnetic field was not beingmeasured routinely. Fortunately, there is another data set, the record of auroras, allowing us to find thedifferences between the CGC as seen in the sunspots and the solar wind. During the Dalton minimumalmost all auroras seen in Northern Sweden appeared in the far north indicating that the interplanetary fieldand solar polar field were also low [Feynman and Silverman, 1980]. These conditions continued for severaldecades around each XSM. All of the CGC minima since 1700 are characterized by a sunspot cycle with anamplitude less than 70, and an extended solar cycle length. It has also been estimated that each of theseCGC minima had a diminished total solar irradiance [Krivova et al., 2007; Kopp and Lean, 2010].
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Figure 5. (a) The annual averaged aa index in 1868–2013. The data are from http://ngdc.noaa.gov/stp/geomagnetic.data. (b) Black circles mark the 21st century aa index. The asterisks and pluses mark the nineteenth to twentieth centuryaa index and the IHV index, correspondingly (see text).
As we noted above, the CGC has a very different appearance in the annual average aa index (Figure 5) thanin the sunspot number (Figure 1). The aa at SSN minima between 1900 and 1954 increased almost linearlyindicating a systematic change in the solar wind [Feynman and Crooker, 1978]. The increase was so largethat the annual aa at solar maximum in 1910 is about the same as the 1954 aa minimum. To under-stand this difference in behavior of SSN and aa we need to look at the solar wind parameters that the aaactually measures.
The estimation of the solar wind parameters expected at Earth from the magnetic fields measured on theSun is done by using the observations of the photospheric fields and extending them into space. Recently,Riley [2007] has reviewed the process developed by Schatten et al. [1969], Wang and Sheeley [1995], and oth-ers of calculating the magnetic field in the solar wind at 1 AU from the measured fields at the photosphereusing the traditional procedure based on the existence of Potential Field Source Surface (PFSS) to find thefield that would exist at a source surface due to the measured photospheric field modeled within the Sun’satmosphere in the absence of currents. For a recent review of the solar coronal magnetic field see Mackayand Yeates [2012]. The source surface is usually assumed to be at about 2.5 R𝜐. It is then assumed that fieldlines that cross that surface are perpendicular to it, i.e., are the value of the Br component, which is frozeninto the plasma and carried to Earth. This model is called “the open flux approximation” [Wang and Sheeley,1995; Lockwood, 2013]. Riley [2007] tested this model against the measured flux at 1 AU and found that themodeled and observed fluxes do not agree except at solar minimum. He then investigated the relationshipbetween the annual averaged sunspot number and the annual number of CMEs to show a 90% correla-tion between these quantities [see also Feynman and Gu, 1986] and estimated that each interplanetary CME
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(ICME) dominated the solar wind at Earth for about 2 days. For the sunspot maxima in the first four cycles(shown in Figure 3) ICMEs would dominate the solar wind between one third and two thirds of the time dur-ing those years. Riley found a very much improved correspondence between the observed fields at 1 AUand his modeled fields when he included additional flux contribution associated with the ICMEs and propor-tional to the annual sunspot number. This step is justified because the PFSS (Potential Field Source SurfaceModels) and similar models based on the use of synoptic magnetic maps, which have 27 day cadence, missthe magnetic field related to the intermittently erupting CMEs.
The aa index of geomagnetic activity at Earth reflects this distinction in that there are two components ofactivity, one of them carries information about the solar toroidal field erupting at the solar surface (relatedto sunspots, flares, and CMEs), the other carries the information about the poloidal field (related to coronalholes). This splitting is justified by earlier studies that have shown that there are two components of the fastsolar wind (transient and quasi-stationary) and two clearly different types of geomagnetic storms: suddencommencement and recurrent storms. The sudden commencement storm is caused by CMEs (associatedwith toroidal magnetic field at the Sun) and the recurrent storms caused by corotating interaction regions(associated with the poloidal field at the Sun). While the aa index shows an 11 year variation, its correla-tion to the annual sunspot number is less than 0.5 [Feynman and Gu, 1986]. The analysis by Feynman andCrooker [1978] showed that the residual of the aa index (after subtraction of the component proportionalto the sunspot number) is strongly related to recurrent geomagnetic storms thus representing the activitycoming from coronal holes. Ruzmaikin and Feynman [2001], see also Georgieva and Kirov [2011], associatedthese components of the aa index with the toroidal and poloidal components of the solar magnetic field,respectively, and introduced the notation aaT and aaP :
aaT = 0.7 SSN + min(aa), aaP = aa − aaT .
The coefficients of this linear dependence between the aaT and SSN were determined by the minimalline above which the values of aa index lie in the SSN-aa scattergram [see Ruzmaikin and Feynman, 2001,Figure 2]. The precise values of the minimum level min(aa) is not important; it slightly depends on thelength of the time series and was calculated as 5.17 nT in Ruzmaikin and Feynman [2001].
Figure 6 shows the time series of these two components of the aa index since 1868 (Figure 6a) and theirphase difference (Figure 6b). We see that the phase shift between the two components of aa index followsthe CGC. During the first half of the CGC (from 1901 to 1951) the phase changed by about π, i.e., the relation-ship between the components changed from being in phase to an antiphase state. It has almost returnedto the in-phase state at present. Note that this long-term phase dependence is not sensitive to the empiricalcoefficients used in the aa dependence on SSN. The phase shift between the poloidal and toroidal com-ponents of the solar magnetic field is necessarily present in a dynamo process. In kinematic dynamos it isdetermined by the ratio of two basic sources of the mean-field generation: differential rotation and meanhelicity [Stix, 1976]. Ruzmaikin and Feynman [2001] compares the phase shift obtained from the aa indexand the prediction of kinematic dynamos. A potential role of this phase shift in the context of nonlineardynamos will be discussed below in section 6.
The solar wind drives aurora in the Earth’s magnetosphere. The aurora occurs in an oval around the geo-magnetic poles. When the magnetosphere interacts with a solar wind having a high flow pressure and moreimportantly an enhanced amount of open magnetospheric flux, the aa index increases, and the auroral ovalexpands and moves equatorward so that aurora can be observed at middle and low latitudes. The aa indexand the number of auroras seen in all of Sweden have a correlation of 0.97 for the 9 years when both datasets are available (1868–1876).
The CGC minima are seen most clearly in the observed absence of midlatitude auroras [Silverman andFeynman, 1980; Silverman, 1992]. Because midlatitude auroras are so intermittent and spectacular, thereare records of them from both Europe and China. Studies of these records have resulted in scientificallyuseful data from 450 A.D. to 1450 A.D. [Siscoe, 1980]. This data set clearly indicates that midlatitude auroraexhibited about a 90 year period in frequency of their occurrence [Silverman and Feynman, 1980; Feynmanand Fougere, 1984]. The solar cycle just before the 1905 XSM had a maximum SSN of 63. There was a sharpdecline in the number of midlatitude aurora observed during that CGC minimum [Silverman, 1992]. Duringthe 21st century XSM very few aurora were observed even as far north as the observing station in Kiruna,Sweden (cf. archives of all-sky camera data www.irf.se/allsky).
Figure 6. (a) The monthly averaged aa index 1868–2013 split into two parts as explained in the text and filtered withthe 32 month running average. The rise and fall of both components is in accord with the recent CGC (1900–2012).(b) The phase difference between the two time series shown in Figure 6a. It is calculated by taking the differencebetween the maxima of these two time series and divided by the half of the period found using the solid curve. Thisnormalization expresses the phase difference in units of π.
To illustrate the effect of the increase in the geomagnetic activity on auroral observations we have drawn aline in Figure 5 at aa = 23. If midlatitude aurora were only seen when the aa exceeded this line, then almostno midlatitude aurora would be seen between 1870 and 1925, but they would be common between 1937and 2004. If this pattern of geomagnetic activity has repeated every 90–100 years, then midlatitude aurorawould periodically appear. In fact, midlatitude and low-latitude aurora were very rarely seen in either theSouthern Sweden or the Boston New Haven area of New England for 40 years, between 1795 and 1835,but they were very common before and after that time [Feynman and Silverman, 1980], illustrating why themidlatitude auroras reflect the CGC more clearly than the aa index.
5. Summary of Findings That the CGC Minimum is Consistent With the XSM
Here we summarize the observational findings evidencing that the recent 23/24 cycle XSM is a minimum ofa Centennial Gleissberg Cycle. This XSM, in turn, confirms the existence of the CGC as a centennial mode ofsolar variability. It completes the CGC that started in the beginning of the twentieth century for which thereis a rich set of solar, solar wind, and geomagnetic data. The analysis of these data indicates that the recentXSM is a typical CGC minimum. The long-term centennial variation of the amplitude and period of the solarcycle shows systematic extended minima in the beginning of several centuries, including the current 21stcentury (Table 1). The CGC is a real solar variation that has been established as operating 80% of the time inthe last 1500 years [Feynman and Ruzmaikin, 2011] and probably for 9400 years as shown in records of 10Beand 14C [McCracken et al., 2013].
The detailed data sets for the past century allow us to identify specific features of the CGC that can help tounderstand its origin, particularly the following:
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Table 1. CGC Identified 80% of the Time in the Last 1500 Years
When Time Period Name Reference
1000 years 450–1450 A.D. Auroral CGCs Siscoe [1980] and Feynman and Fougere [1984]
18th century 1710–1720 End of MM Eddy [1976]
19th century 1790–1830 Dalton Min Schove [1983]
20th century 1910 Gleissberg Min Feynman and Crooker [1978]
21st century 2006–2011 Current Min Silverman [1992]
1. The phase difference between the proxies for toroidal and poloidal components of the solar magneticfield changes as an integral part of the CGC. The common notion that poloidal magnetic field maximizesat solar minima is not exact.
2. There is a change in the solar surface rotation rate on the centennial time scale, as indicated by themotions of sunspots [Pulkkinen and Tuominen, 1998; Javaraiah et al., 2005]. Since the surface rotation dis-plays the radial-latitudinal profile of the solar differential rotation in the convection zone [Schou, 1998],these changes are expected to reflect the changes of the whole profile. The dynamo modeling [Tobias,1996; Pipin, 1999] (see the next section) showed that the changes in the differential rotation in the solarconvection zone can be caused by the action of the dynamo generated magnetic fields.
3. The dipole and quadrupole components of the solar magnetic field change in relative amplitude andphase. The relative amplitude of the quadrupole mode becomes larger as the CGC approaches its mini-mum. The dipole and quadrupole oscillate with slightly different periods indicating beats at a centennialperiod. The sum of the dipole and quadrupole indicates a north-south asymmetry [Pulkkinen et al., 1999;Zolotova et al., 2009].
4. The joint probability distribution of the solar wind flow pressure and magnetic field undergo a systematicchange in accord with the centennial solar variability.
6. Discussion: On Possible Origin of CGC
The solar cycle exhibits an amplitude and possibly frequency modulation on the centennial time scale(Figure 1). The frequency (phase) modulation is also reflected in a change in the phase difference of thepoloidal and toroidal components of the solar magnetic field (Figure 6). Note that the centennial modula-tion is not a linear harmonic function but includes a band of frequencies. The first thing the theory has todo is to explain the origin of the centennial time scale. It also has to explain the observed phase changebetween the toroidal and poloidal magnetic field components and the time dependence of the dipoleand quadrupole components of the poloidal magnetic field, as indicated by observations discussed in theprevious sections.
Since solar variability is driven by the magnetohydrodynamics of the solar convection zone, the explana-tion of the CGC has to be found in the solar dynamo. The standard dynamo is the process by which themean magnetic field is generated by the differential rotation, mean helicity, and meridional flow and isdamped by the turbulent resistivity of convective motions. (For the foundations of dynamo theory seeMoffatt [1978], Parker [1979], Krause and Rädler [1980], and Zeldovich et al. [1984]. For current develop-ments see the living review by Charbonneau [2010], who gives a critical assessment of the solar dynamomodels.) These sources of the magnetic field generation and damping were assumed as given quantitiesand usually estimated from helioseismic observations or from modeling. Many solar observers favor theso-called Babcock-Leighton mechanism of reproduction of the poloidal magnetic field from the decay ofbipolar active regions. At the heart of this mechanism is a systematic tilt of the emerging active regionswith respect to the east-west direction on the solar surface. The tilt angle is a critical part of the source ofthe poloidal field. Dasi-Espuig et al. [2010], by carefully analyzing the sunspots data from Mount Wilson andKodaikanal observatories, have found an anticorrelation between the mean tilt angle of a given cycle andthe amplitude of the cycle and a good correlation between the source term of the Babcock-Leighton modeland the amplitude of the next solar cycle. Note that the source term in the Babcock-Leighton model wasnot derived from the first principles but introduced only on observational grounds by the two outstand-ing solar observers, Babcock and Leighton. Thus, it remains unclear whether the tilt is specific only to theBabcock-Leighton dynamo or present in the dynamos derived from the first principles in the mean-fieldmagnetohydrodynamic approximation.
Journal of Geophysical Research: Space Physics 10.1002/2013JA019478
It is not the goal of our paper to discuss which dynamos are better designed to simulate the 11 yearsolar cycle. (For this purpose we refer a reader to the review by Charbonneau [2010].) We touch hereonly the known aspects of dynamo mechanisms that could be relevant to the centennial variability withan intention to stimulate the development of dynamo models that embrace the long-term variations.To address the observed centennial changes the dynamo must be substantially nonlinear to take intoaccount the evolution of the main modes of the magnetic field and the back action of the magnetic fieldon the differential rotation and mean helicity, meridional circulation, and possibly the source term in theBabcock-Leighton mechanism.
Early numerical simulations in the kinematic approximation (i.e., with the fixed differential rotation andmean helicity), which included the two main modes of the field (dipole and quadrupole), showed thatthere is a solution in which the quadrupole oscillates with a period that is slightly different from the 11 yearoscillation of the main dipole mode [Ivanova and Ruzmaikin, 1976]. The beats between the dipole andquadrupole modes oscillating with close periods produce a long-term amplitude modulation. Indeed, whenTq = Td(1−𝜖), where Td and Tq are the oscillation periods of the dipole and quadruple modes and 𝜖 is a smallnumber (0.1); the modulation period 𝜏 is
1𝜏= 1
Tq− 1
Td, 𝜏 =
Td
𝜖=
11 years0.1
= 100 years.
The sum of the two modes produces a north-south asymmetric magnetic field since the dipole has theopposite and quadrupole the same signs of the field at the poles.
The situation is more complicated in the nonlinear dynamo regime when the stronger dipole mode maycapture the oscillation of the weaker quadrupole mode leading to the field oscillating with Td period.The conditions for the capture and nonlinear beats regimes were investigated in a simplified model byKleeorin and Ruzmaikin [1984]. Brandenburg et al. [1989] numerically found stable solutions of even, odd, andmixed north-south parity modes, some lie on a two-periodic torus thus adding a long period. Tobias et al.[1995] considered modes of a fixed parity (dipole or quadrupole) to show how the interaction of magneticand velocity fields produce quasiperiodic modulations. At this time observations cannot detect statisticallysignificant beats between the dipole and quadrupole modes (Figure 2). The observations only show thedifferent amplitude changes of these two modes that must be explained by theory.
Hathaway and Rightmire [2010] suggested that the cause of the current deep minimum is related to changesin the meridional flow that carries magnetic elements of one polarity to the poles. This process leads toannihilation of old polar fields and produces polar fields of the opposite polarity. A fast meridional flowinhibits cancelation of fields of opposite polarities across the solar equator thus carrying elements of bothpolarities to the poles and leading to a longer time to reverse the old polar fields. Kinematic dynamo simu-lations [Nandy et al., 2011] demonstrated that a fast meridional flow in the first half of a solar cycle, followedby a slower flow in the second half of the cycle, reproduces observed characteristics of the minimum ofsunspot cycle 23. The model predicts that, in general, very deep minima are associated with weak polarfields. However, kinematic models (with prescribed changes in meridional flow) do not provide estimates ofthe long-term periodicity and could not explain why the meridional flow is changing. For this purpose oneneeds to consider nonlinear dynamos.
Some insights into the origin of the CGC can be found from estimates of the magnetic field feedbackon the solar differential rotation and mean helicity. The nonlinear dynamo equations can be written in afollowing form:
𝜕Bp
𝜕t= rot(𝛼Be𝝓) + ΔBp, (1)
𝜕B𝜕t
= D𝛁ΩBp + ΔB, (2)
𝜕𝛼m
𝜕t= N(Bp, B) −
𝛼m
𝜏, (3)
𝜕Ω𝜕t
= ∇iTi𝜙, (4)
where Bp is the poloidal magnetic field, B is the toroidal magnetic field, D = 𝛼0Ω0R3∕𝜂2T is a dimensionless
dynamo number (𝛼0 and Ω0 are characteristic values of the differential rotation and mean helicity, R is the
Journal of Geophysical Research: Space Physics 10.1002/2013JA019478
solar radius, and 𝜂T is a turbulent diffusivity in the convection zone), N is a nonlinear, second-order operator,and Tij is the stress tensor that includes the Reynolds (hydrodynamic) and Maxwell (magnetic) stresses. Themean helicity is split into a kinematic part, and the part influenced by the magnetic field 𝛼 = 𝛼k + 𝛼m.
The models suggested by Tobias [1996] and Pipin [1999] interpret the centennial variability as resulting fromthe magnetic feedback on the angular momentum fluxes that maintain the radial-latitudinal profile of dif-ferential rotation in the solar convection zone, see equation (4). The period of the long-term variation isidentified as a relaxation time needed to reestablish the profile of the differential rotation after a pertur-bation of the angular momentum by the magnetic torque. Pipin [1999] estimated the relaxation time as5R2∕𝜂T ≈100 years, where R = 7 × 1010 cm is the solar radius and 𝜂T ≈ 1013 cm2/s is the turbulent diffusivityin the convection zone.
Kleeorin et al. [1994] analyzed the back action of the magnetic field on the mean helicity, see equation (3),and estimated the relaxation time as 𝜏 = l2∕8𝜋2𝜂 ≈ 100 years, where l ≈ 1.5 × 109 cm is a mixing length ofthe turbulent convection and 𝜂 ≈ 107 cm2/s is the magnetic plasma diffusivity in the convection zone.
These theoretical investigations suggest physical processes that may be operating in the solar convectionzone on centennial time scales. The ratio of the amplitudes of poloidal and toroidal components of the solarmagnetic field and the time phase shift between them is determined by the ratio of their generation sources[Stix, 1976; Bigazzi and Ruzmaikin, 2004]. To understand the change we note that the solution of the dynamoequations have the form of waves propagating in the direction of the wave vector k
B ∝ exp[i(𝜔t + kr + 𝜙)], Bp ∝ exp[i(𝜔t + kr)],
where 𝜔2 = krR𝛼|∇Ω|∕2.
According to a numerical investigation in a simple kinematic dynamo model [Stix, 1976] and in a moremodern dynamo model, which includes the helioseismic reconstruction of the differential rotation [Bigazziand Ruzmaikin, 2004], the phase difference between the poloidal and toroidal components depends onthe ratio 𝜔∕k|∇Ω| inside the solar convection zone. The observed phase shift of about π between thetoroidal and poloidal components from the Gleissberg minimum (about 1910) to the Gleissberg maximum(circa 1965–1970) proxied by the aa index (Figure 6) constrains this parameter of the solar dynamo. It alsosupports the suggestion that the mean helicity and/or differential rotation are changing on centennialtime scale.
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AcknowledgmentsWe thank reviewers for helpful, criticalcomments. This work was supportedin part by the Jet Propulsion Lab-oratory of the California Instituteof Technology, under a contractwith the National Aeronautics andSpace Administration.
Yuming Wang thanks the review-ers for their assistance in evaluatingthis paper.
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