Internal-cycle variation of solar differential rotation
aa r X i v : . [ a s t r o - ph . S R ] M a y INTERNAL-CYCLE VARIATION OF SOLAR DIFFERENTIALROTATION
K. J. LI , , J. L. XIE , , X. J. SHI , National Astronomical Observatories/Yunnan Observatory, CAS, Kunming650011, China Key Laboratory of Solar Activity, National Astronomical Observatories,CAS, Beijing 100012, China Graduate School of CAS, Beijing 100863, China
Abstrct.
The latitudinal distributions of the yearly mean rotation ratesmeasured respectively by Suzuki in 1998 and 2012 and Pulkkinen & Tuomi-nen in 1998 are utilized to investigate internal-cycle variation of solar dif-ferential rotation. The rotation rate at the solar Equator seems to decreasesince cycle 10 onwards. The coefficient B of solar differential rotation, whichrepresents the latitudinal gradient of rotation, is found smaller in the severalyears after the minimum of a solar cycle than in the several years after themaximum time of the cycle, and it peaks several years after the maximumtime of the solar cycle. The internal-cycle variation of the solar rotationrates looks similar in profile to that of the coefficient B . A new explanationis proposed to address such a solar-cycle related variation of the solar rotationrates. Weak magnetic fields may more effectively reflect differentiation at lowlatitudes with high rotation rates than at high latitudes with low rotationrates, and strong magnetic fields may more effectively repress differentiationat relatively low latitudes than at high latitudes. The internal-cycle varia-tion is inferred to the result of both the latitudinal migration of the surfacetorsional pattern and the repression of strong magnetic activity to differen-tiation. Sun: rotation– Sun: activity– Sun: sunspots
The Sun’s atmosphere is found to rotate faster at the equatorial region thanat higher latitude regions. In a specific word, it rotates in a circular courseby 26 days at the solar Equator but 30 days at 60 ◦ latitude, which is the so-called differential rotation (Balthasar & W¨ o hl 1980; Gilman & Howard 1984;1heeley, Wang & Nash 1992; Rybak 1994; Altrock 2003; Song & Wang 2005;Chu et al. 2010; W¨ o hl et al. 2010; Li et al. 2013). Two main methods havebeen usually exploited to measure rotation velocity of the solar atmosphere:the tracer method and the spectroscopicmethod (Howard, Gilman & Gilman 1984; Pulkkinen & Tuominen 1998a;Braj˘ s a et al. 2000, 2002; W¨ o hl & Schmidt 2000; Le Mouel et al. 2007; Liet al. 2012). In the solar interior, solar rotation rate is also determined butby a specialized method: the helioseismology measurement method (Howeet al. 2000a, 2000b; Antia & Basu 2001), and the latitudinal migrationof rotation angular velocity in solar interior (Howe et al. 2009) is foundsimilar to the torsional oscillation pattern of the solar atmosphere measuredby the spectroscopic method (Howard & LaBonte 1980; LaBonte & Howard1982; Schr¨ o ter 1985). At the present, observations and studies for the solardifferential rotation have taken a great achievement (Howard 1984; Schr¨ o ter1985; Snodgrass 1992; Paterno 2010; Li et al. 2012) However, there are stillsome aspects, for example, the solar-cycle related and long-term variationsof solar rotation rate, unknown now (Komm, Howard & Harvey 1993; Ulrich& Bertello 1996; Stix 2002; Li et al., 2011a, 2011b).In this study, we will investigate internal-cycle variation of solar differen-tial rotation, using the measurements of rotation rates made by Suzuki (1998,2012) and Pulkkinen & Tuominen (1998a). A new explanation is proposedto address such a solar-cycle related variation of the solar rotation rates. Daily photographic observations of sunspots at the solar full disk have beenmade with a refractor of 102 mm aperture and 1200 mm focal length bySuzuki since the year of 1988, and the positions of sunspot groups at the solardisk are obtained (Suzuki 1998, 2012). He measured annual mean rotationrates of sunspots in 5 latitude bin during cycles 22 to 23 (from 1998 to 2006)through analyzing his observational data of sunspots. The measurements are2iven in the Table 1 of Suzuki (1998) for the years of 1988 to 1995 and inthe Table 1 of Suzuki (2006) for the years of 1996 to 2006.The solar differential rotation is usually expressed by the standard for-mula (Newton & Nunn 1951): ω ( φ ) = A + Bsin φ where ω ( φ ) is the solar sidereal angular velocity at latitude φ and the coeffi-cients A and B represent the equatorial rotation rate and the latitudinal gra-dient of the rotation, respectively (Howard 1984). The latitudinal distribu-tion of annual mean sidereal rotation rates measured by Suzuki (1998, 2012)is fitted by the formula for each year. Figure 1 shows the cross-correlationcoefficient of the formula fitting to the latitudinal distribution of the annualmean rotation rates, and the corresponding tabulated value at the 95% confi-dence level is also given. The calculated correlation coefficient is larger thanthe corresponding tabulated value for all years except the years of 1993 and1994, indicating that the formula can statistically significantly give a goodfitting to the Suzuki’s measurements of annual mean rotation rates. In thesetwo years (1993 and 1994), the correlation coefficients are both less than thecorresponding tabulated values at the 95% confidence level. Thus for thesetwo years, the fitting values of A and B are replaced by the linearly extrap-olated values of their neighboring two points. Figure 2 shows the obtainedcoefficients A and B . A linear fitting is taken to the coefficient A varyingwith time t (in years), and resultantly, A = 39 . − . × t , and thecorrelation coefficient is 0.4246, which is statistically significant at the 92%confidence level. There is a decrease trend for A , and the decrease rate is0 . /day per year. A 5-point smoothing is taken to the coefficient B ,which is shown in the figure. A special feature for B is that its absolutevalue is larger in the several years after the minimum of a solar cycle than inthe several years after the maximum time of the cycle, and the absolute valueof B clearly decreases and then increases within the falling part of a sunspotcycle. As the figure shows, a short-term effect can be seen for B varyingwithin a solar cycle, and a long-term effect, for A trending to decrease. Thecorrelation coefficient of yearly sunspot numbers with B is calculated to be0.521, which is statistically significant at the confidential level of 95%. Whilethe correlation coefficient of yearly sunspot numbers with A is -0.384, whichis of no significance. For the solar surface rotation rate at the solar Equator(the coefficient A ), there exists a secular decrease of statistical significance3ven since Cycle 12 onwards (Javaraiah, Bertello & Ulrich 2005a, 2005b; Liet al. 2013).Based on the obtained coefficients A and B , we calculate latitudinal dis-tribution of rotation rates in each year of 1988 to 2006 through the standardformula, which is shown in Figure 3. Rotation rates are found to obvi-ously show a migration within a solar cycle, but such a migration is differentat the falling part of a solar cycle from both the latitudinal migration ofsunspots and the torsional oscillation pattern of solar surface differential ro-tation (Snodgrass 1987; Li et al. 2008). Figure 4 show four isopleth lines ofrotation rates and their corresponding 5-point smoothing lines. The isoplethvalues of rotation rates are 14.4, 14.2, 14.0, and 13 . o day − in turn from lowto high latitudes. Shown also in the figure are the minimum and maximumtimes of sunspot cycles. As Figures 3 and 4 show, the migration of rotationrates seems thwarted at the falling phase of a solar cycle, compared with theoscillation pattern of solar surface differential rotation (Snodgrass 1987; Liet al. 2008). It is inferred that strong magnetic fields should repress solardifferential rotation, in agreement with Braj˘ s a et al. (2006) and W¨ o hl et al.(2010). & Tuominen (1998a)
The Royal Observatory in Greenwich started sunspot observations, the long-lasting Greenwich Photoheliographic Results (GPR) in 1874, lasting about103 years, and it stopped in 1976. After that, one of the still continuingrecords of sunspots has been made by the Solar Optical Observing Network(SOON) of the US Air Force together with the US National Oceanic and At-mospheric Administration (NOAA) (Pulkkinen & Tuominen 1998a). TheseGPR and SOON/NOAA data sets can be found at the internet web site .Pulkkinen & Tuominen(1998a) used these GPR and SOON/NOAA data incycles 10 to 22 namely at the time interval of 1874 to 1996 to study velocitystructures from sunspot statistics. They gave in their Figure 3 the rotationprofiles of the data at the equatorial range between latitudes ±
20 degreesat different phases of cycle. These rotational velocity values are the siderealones. The figure is 3-times enlarged, then all points of the rotation profiles in .
5% confidence level. The calculated correlation coefficients are larger thanthe corresponding tabulated values for all years, indicating that the formulacan statistically significantly give a good fitting to the annual rotation profile.The correlation coefficient is lowest in and around solar activity minima dueto the lower number of sunspots on the Sun in those time intervals. Figure6 shows the obtained coefficients A and B at different phases of a Schwabecycle. A linear fitting is taken to the coefficient A varying with time t of asolar cycle, and resultantly, A = 14 . − . × t , and the correlationcoefficient is 0.6029, which is statistically significant at the 93% confidencelevel. There is a decrease trend for the coefficient A , and the decrease rateis 0 . /day per year. A 3-point smoothing is taken to the coefficient B ,which is shown in the figure. The same special feature for B is obtained asthat obtained through analyzing the data of Suzuki (1998, 2012).Based on the fitting values of the coefficients A and B , we also calculatelatitudinal distribution of rotation rates in each year of a solar cycle throughthe standard formula, which is shown in Figure 7. Rotation rates are found toobviously show the same migration within a solar cycle as mentioned above.Figure 8 shows four isopleth lines of rotation rates and their corresponding3-point smoothing lines. The isopleth values of rotation rates are 14.2, 13.9,13.6, and 13 . o day − in turn from low to high latitudes. As Figures 7and 8 show, large sunspots seem to hinder the torsional oscillation patternshifting towards the Equator 1 ∼ CONCLUSIONS AND DISCUSSION
The latitudinal distributions of the yearly mean rotation rates measuredrespectively by Suzuki (1998, 2012) and Pulkkinen & Tuominen (1998a) areexploited to investigate internal-cycle variation of solar differential rotation.Firstly , they are fitted by the standard formula of solar differential rotation.Resultantly, the rotation rate at the solar Equator is then found to decreasesince cycle 10 onwards, and the decrease rate is about 0 . /day per yearwithin a solar cycle. For the coefficient B , its absolute value is found largerin the several years after the minimum of a sunspot cycle than in the severalyears after the maximum time of the cycle, and the absolute value clearlydecreases and then increases within the falling part of the sunspot cycle,namely, the coefficient B peaks several years after the maximum time ofthe solar cycle. Such a profile of the coefficient B in a solar cycle was alsogiven in the Figure 5 of Javaraiah (2003). Although the variations of thecoefficients A and B within a solar cycle obtained through analyzing the dataof Pulkkinen & Tuominen (1998a) are similar to those through analyzing thedata of Suzuki (1998, 2012), the former should be more plausible, becausethe data utilized by Pulkkinen & Tuominen (1998a) are more reliable inobservations and much longer in time than by Suzuki (1998, 2012). Thus atfollows we mainly focus attention on the former.The differential rotation of solar atmosphere has a periodical pattern ofchange. Such a pattern can be described by the so-called torsional oscilla-tion, in which the solar differential rotation should be cyclically speeded upor slowed down in certain zones of latitude while elsewhere the rotation re-mains essentially steady (Snodgrass & Howard 1985; Li et al. 2008; Li et al.2012). The zones of anomalous rotation move on the Sun in wavelike fashion,keeping pace with and flanking the zones of magnetic activity (LaBonte &Howard, 1982; Snodgrass & Howard, 1985). The surface torsional pattern,and perhaps the magnetic activity as well, are only the shadows of anotherunknown phenomenon occurring within the convection zone (Snodgrass 1987;Li et al. 2008). In this investigation, the internal-cycle variation of solar dif-ferential rotation can be explained as follows on the framework of the surfacetorsional pattern, which is briefly stated above, and by the magnetic activitytogether.Braj˘ s a, Ru˘ z djak & W¨ o hl (2006) once investigated solar-cycle related vari-ations of solar rotation rate. They found a higher than average rotation6elocity in the minimum time of a Schwabe cycle, and a plausible interpre-tation was then given. When magnetic fields are weaker, one can expect amore pronounced differential rotation yielding a higher rotation velocity atlow latitudes on an average (Braj˘ s a, Ru˘ z djak & W¨ o hl 2006). As Figures 2and 6 shows, more pronounced differentiation of rotation rates appears in-deed at the first 4 years of a solar cycle than at at the second 4 years, dueto that weaker magnetic fields should appear at the first 4 years. Strongmagnetic fields should repress differentiation, but weak magnetic fields seemto just reflect differentiation of rotation rates. Further, weak magnetic fieldsmay more effectively reflect differentiation at low latitudes with high rota-tion rates than at high latitudes with low rotation rates, and strong magneticfields may more effectively repress differentiation at relatively low latitudesthan at high latitudes. As Figures 2 and 6 display, the coefficient B maybe divided into three parts. Part one spans from the start to the 4 th yearof a solar cycle, the absolute B is approximately a constant or slightly fluc-tuates. Relatively high latitudes and relatively weak magnetic fields at thistime interval make the repression action of sunspots less obvious than afterthis interval. Part two spans from the 4 th to the 7 th year, the absolute B decreases. When solar activity is progressing into this part, sunspots appearat lower and lower latitudes, magnetic fields repress differentiation more andmore effectively, and differentiation appears less and less conspicuously, thusthe absolute B decreases within this part. Part three spans from the 7 th year to the end of a solar cycle. Within this part, magnetic fields becomemore and more weak, they repress differentiation less and less effectively,and sunspots appearing at more and more low latitudes lead to that thedifferentiation reflected by latitudinal migration should be more and moreconspicuous, thus, the absolute B increases (Li et al. 2012). In sum, theinternal-cycle variation of solar differential rotation is inferred to the resultof both the latitudinal migration of the surface torsional pattern and therepression of strong magnetic activity. It means that measurements of differ-ential rotation should different at different phase of a Schwabe cycle or/andat different latitudes (different spacial positions of observed objects on thesolar disk), that is the main reason why too many different results aboutsolar differential rotation exist at the present (Howard 1984; Schr¨ o ter 1985;Snodgrass 1992; Beck 1999; Paterno 2010; Li et al. 2012).The solar differential rotation is not a fossil but is proposed generated andcontinuously maintained by the angular momentum transport from higher7atitudes toward the Equator (Pulkkinen & Tuominen 1998a). Measurementsfrom the GPR by Ward (1965) revealed the existence of this transport, mostlyby the Reynolds stress. Vr˘ s nak et al. (2003) found a statistically signifi-cant correlation between rotation residuals and meridional motions, throughtracing “point-like structures” (predominantly young coronal bright points),indicating that the existence of the equatorward transport of angular mo-mentum. Tracing of coronal bright points (CBPs) provides an extension ofthe Reynolds stress analysis to high latitudes which could be an importanttool to investigate the dependence of the Reynolds stress on the latitude(Vr˘ s nak et al. 2003), because CBPs are interesting features to be used forthe rotation estimation along a solar cycle since they appear even on the fulldisk and at both minimum and maximum cycle phases (Zaatri et al. 2009;W¨ o hl et al. 2010). The torsional oscillation that was found by Howard andLaBonte (1980) indicated that latitudinal motions should exist, and theymay result both from hydrodynamic circulation and solar magnetic cycle,therefore the Reynolds stress is strongly present (Tuominen, Tuominen &Kyr¨ o l¨ a inen 1983; R¨ u diger et al. 1986; Pulkkinen et al. 1993; Pulkkinen& Tuominen 1998a). The horizontal Reynolds stress should be a function(dependence) of not only the gradients of rotation but also rotation itself(Pulkkinen & Tuominen 1998a). We propose here that the spacial variationsof the solar magnetic activity act on differential rotations in different ways(see the explanation on Figures 2 and 6). It means that the latitudinal shiftvelocity should be related with differential rotations, accordingly supportingthis dependence.Comparison of Figure 8 with Figure 6 shows that the internal-cycle varia-tion (isopleth lines) of the solar rotation rates looks similar in profile to thatof the coefficient B , and the former is mainly reflected by the latter. Thestrong magnetic fields should hinder the torsional oscillation pattern migrat-ing towards the Equator, and a block is thus formed in the migration “river”.The block peaks when the B peaks. The internal-cycle variation of the solarrotation rates is inferred to the result of both the latitudinal migration of thesurface torsional pattern and the repression of strong magnetic activity todifferentiation. Indeed the surface torsional pattern, and perhaps the mag-netic activity as well, are only the shadows of another unknown phenomenonoccurring within the convection zone (Snodgrass 1987; Li et al. 2008).Based on the above explanation about the variation profile of B withina solar cycle, the temporal distribution of magnetic activity strength in a8olar cycle should give rise to B periodically distributed in a solar cycle, butthe spacial distribution of magnetic activity strength should disturb B toform such a solar-cycle-period distribution. Therefore, the variation profileof the coefficient B in a solar cycle (see Figures 2 and 6) is similar to butobviously different from the variation profile of sunspot numbers in a solarcycle. This seemingly implies that the coefficient B should possibly have anot-too-strong or weak relation with sunspot activity at the scale of solarcycles. We calculated the correlation coefficient of yearly sunspot numbersrespectively with the coefficients A and B determined through the data ofSuzuki (1998, 2012), and resultantly it is 0.521 for B , which is statisticallysignificant at the confidential level of 95%. But for A it is -0.384, whichis of no significance. However, Jurdana- ˘ S epi´ c et al. (2011) recently foundthat the coefficient A is significantly related with sunspot activity, while B isnot related, through tracing small bright coronal structures. Chandra et al.(2010) found that B should not show any systematic variations for the softX-ray corona, which is in agreement with the resutls of Jurdana- ˘ S epi´ c et al.(2011). The correlation of sunspot activity with the rotation coefficients de-termined by tracing sunspots contradicts that determined by tracing coronalactivity events. The possible reason is inferred to be that the coronal mag-netic field is much weaker than the sunspot magnetic field. For the time scalelonger than solar cycles, the coefficient A is found negatively correlated withsunspot activity, while B should be hardly correlated (Javaraiah, Bertello &Ulrich 2005a, 2005b; Javaraiah & Ulrich 2006; Chandra & Vats 2011; Li etal. 2012).We thank the anonymous referees for their careful reading of the manuscriptand constructive comments which improved the original version of the manuscript.This work is supported by the 973 programs 2012CB957801 and 2011CB811406,the National Natural Science Foundation of China (11273057, 11221063,11147125, and 11073010), and the Chinese Academy of Sciences.9 eferences [1] Altrock, R. C. 2003, Sol. Phys., 213, 23[2] Antia, H. M., & Basu S. 2001, ApJ, 559, L67[3] Balthasar, H., & W¨ o hl, H. 1980, A & A , 92, 111[4] Braj˘ s a, R., Ru˘ z djak, V., Vr˘ s nak, B., W¨ o hl, H., Pohjolainen, S., & Urpo,S. 2000, Sol. Phys. 196, 279[5] Braj˘ s a, R., Ru˘ z djak, D., & W¨ o hl, H. 2006, Sol. 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990 1995 2000 20050.30.40.50.60.70.80.91 *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x *x
Calendar Year C o rr e l a t i on C oe ff i c i en t Figure.1
The correlation coefficient (the asterisk symbol) of the latitudinaldistribution of annual mean sidereal rotation rate measured by Suzuki(1998, 2012) at each year of 1988 to 2006 fitted by the standard formula ofsolar differential rotation. The cross symbol shows its correspondingtabulated value at the 95% confidence level.13
990 1995 2000 200514.414.614.815 A ( deg r ee s da y − ) B ( deg r ee s da y − ) Calendar Year
Figure.2
The top panel: the coefficient A (the bold solid line) obtainedthrough fitting the latitudinal distribution of annual mean sidereal rotationrates measured by Suzuki (1998, 2012) with the standard formula of solardifferential rotation. The dashed line shows the linear fitting to theobtained coefficient A . The bottom panel: the coefficient B (the bold solidline) obtained through fitting the latitudinal distribution of annual meanrotation rates measured by Suzuki (1998, 2012) with the standard formulaof solar differential rotation. The dashed line shows its 5-point smoothing.In the two panels, the thin vertical dashed/solid lines indicate theminimum/maximum times of sunspot cycles.14 alendar Year La t i t ude ( deg r ee ) Figure.3
Latitudinal distribution of sidereal rotation rates in each year of1988 to 2006 given through fitting the latitudinal distribution of annualmean rotation rates measured by Suzuki (1998, 2012). The unit shown onthe color bar is degrees day − . The fraction numbers at the right side ofthe figure from the top to the bottom correspond to the decimal numbers of14.6, 14.4, 14.2, 14.0, 13.8, 13.6, 13.4, and 13.2 in turn.15
990 1992 1994 1996 1998 2000 2002 2004 2006051015202530354045 La t i t ude ( deg r ee ) Calendar Year
Figure.4
Isopleth lines (the solid curves) of the rotation rates shown inFigure 3 and their corresponding 5-point smoothing lines (the bold dashedcurves). The isopleth rotation rates are 14.4, 14.2, 14.0, and 13 . o day − inturn form low to high latitudes. The thin vertical dashed/solid linesindicate the minimum/maximum times of sunspot cycles.16 * * * * * * * * * * Year from minimum C o rr e l a t i on C oe ff i c i en t Figure.5
The correlation coefficient (the asterisk symbol) of the latitudinaldistribution of annual mean sidereal rotation rate measured by Pulkkinen &Tuominen (1998a) at different phases of cycle fitted by the standardformula of solar differential rotations. The dotted line shows itscorresponding tabulated value at the 98 .
5% confidence level.17 A ( deg r ee s da y − ) B ( deg r ee s da y − ) Year from minimum
Figure.6
The top panel: the coefficient A (the solid line) at differentphases of cycle, which is obtained through fitting the latitudinaldistribution of annual mean sidereal rotation rates measured by Pulkkinen& Tuominen (1998a) with the standard formula of solar differentialrotations. The dashed line shows the linear fitting to the obtainedcoefficient A . The bottom panel: the coefficient B (the solid line) atdifferent phases of cycle, which is obtained through fitting the latitudinaldistribution of annual mean rotation rates measured by Pulkkinen &Tuominen (1998a) with the standard formula of solar differential rotation.The dashed line shows its 3-point smoothing.18 ear from minimum La t i t ude ( deg r ee ) Figure.7
Latitudinal distribution of sidereal rotation rates at differentphases of cycle, which is given through fitting the latitudinal distribution ofannual mean rotation rates measured by Pulkkinen & Tuominen (1998a).The unit shown on the color bar is degrees day − . The fraction numbers atthe right side of the figure from the top to the bottom correspond to thedecimal numbers of 14.4, 14.2, 14.0, 13.8, 13.6, 13.4, 13.2, 13.0, and 12.8 inturn.19 La t i t ude ( deg r ee ) Year from minimum
Figure.8
Isopleth lines (the solid curves) of rotation rates shown in Figure7 and their corresponding 3-point smoothing lines (the dashed lines). Theisopleth rotation rates are 14.2, 13.9, 13.6, and 13 . o day −1