SOLAR CYCLE VARIATIONS OF LARGE SCALE FLOWS IN THE SUN Sarbani Basu Institute for Advanced Study, Olden Lane, Princeton NJ 08540, U. S. A. and Astronomy Department, Yale University, P.O. Box 208101 New Haven, CT 06520-8101 USA 0 H. M. Antia 0 Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, 0 India 2 n February5, 2008 a J Abstract. Using data from the Michelson Doppler Imager (MDI) instrument on board the Solar and Heliospheric Observatory (SOHO), we study the large-scale 7 1 velocity fields in theouterpart of the solar convection zone using thering diagram technique.Weuseobservationsfrom fourdifferenttimestostudypossibletemporal 1 variationsinflowvelocity.Wefinddefinitechangesinboththezonalandmeridional v components of the flows. The amplitude of the zonal flow appears to increase with 4 solar activity and the flow pattern also shifts towards lower latitude with time. 9 2 Keywords: Sun:General – Sun:Interior– Sun:Oscillations – Sun:rotation 1 0 0 0 1. Introduction / h p Ring diagram analysis has been extensively used to infer horizontal - o flows in the outer part of the solar convection zone (Hill 1988; Patro´n r t et al. 1997; Basu, Antia and Tripathy 1999). This technique is based s a on the study of three-dimensional power spectra of solar p-modes on a : part of the solar surface, and can be used to study the variation in flow v i velocity with latitude, longitude and time. The latitudinal variation of X flow velocities in both the rotational and meridional component has r a been extensively studied (Schou and Bogart 1998; Basu et al. 1999; Gonzalez Hernandez et al. 1999) and there is a reasonable agreement between results obtained by different workers. There is some indication that the flow velocity also changes with longitude and time (Patro´n et al. 1998), but it is not clear if some of these variations are due to uncertainties in estimating the velocities or some local influence like the presence of active regions. Since the large scale flows, like the differential rotation and merid- ional flows are expected to play an important role in the functioning of the solar dynamo (Choudhuri, Schussler and Dikpati 1995; Brummell, Hurlburt and Toomre 1998), one may expect some changes in these paper.tex; 5/02/2008; 5:26; p.1 2 S.BasuandH.M.Antia flow patterns over the solar cycle. These variations may give us some information about how the solar dynamo operates. From the splittings of global f-modes Schou (1999) has found variations in the zonal flow pattern, which are similar to the torsional oscillations observed at the solar surface. Thus it appears that these oscillations penetrate into the deeperlayers.Itwouldthusbeinterestingtostudypossiblevariationin these large scale flows with solar activity and how far deep these flows extend. With the high quality data collected by Michelson Doppler Imager(MDI)onboardSOHO,overthelast3yearsithasbeenpossible tostudythesetemporalvariationsandinthisworkweattempttostudy possible temporal variation in large scale flows. For simplicity, we have only considered the longitudinal averages, which contain information about the latitudinal variation in these flows. For this purpose, at each latitude we have summed the spectra obtained for different longitudes to get an average spectrum which has information on the average flow velocity at each latitude. The averaging helps us in improving the reli- ability of results thus allowing us to infer relatively small variations in the flow velocities. We have studied both the rotational and meridional components of the flow velocity. 2. Technique In this work we use the data obtained by the MDI instrument to mea- sure the flow velocities at different epochs. We use four sets of data. Set 1 is data for May/June 1996, Set 2 is for July 1996, Set 3 is for April 1997 and Set 4 is for January/February 1998. These data were taken at various phases of the current solar cycle. The data consist of three dimensional power spectra obtained from full disc Dopplergrams. TheDopplergrams weretaken atacadenceof 1minute.Theareabeing studied was tracked at the surfacerotation rate. To minimize the effect of foreshorteningwehave only useddatafor thecentral meridian.Each powerspectrumwasobtainedfromatimeseriesof1664imagescovering 15◦ in longitude and latitude. Successive spectra are separated by 15◦ in heliographic longitude of the central meridian. For each longitude, we have used 15 spectra centered at latitudes ranging from 52.5◦N to 52.5◦S with a spacing of 7.5◦ in latitude. Set1consistsofspectrafromtwelvelongitudes,startingfromcentral meridian at 105◦ to 15◦ of Carrington Rotation 1909 and 360◦ to 300◦ of rotation 1910. Set 2 consists of twelve longitudes from Carrington rotation 1911, central meridian at 285◦ to 120◦, set 3 contains eight longitudes from Carrington rotation 1921, central meridian at 120◦ to 15◦ and set 4 contains nine longitudes from Carrington rotation 1932, paper.tex; 5/02/2008; 5:26; p.2 SolarcyclevariationsoflargescaleflowsintheSun 3 central meridian at 360◦ to 240◦. The choice of these data sets was dictated by theavailability of dataat thetime this analysis was carried out. To extract the flow velocities and other mode parameters from the three dimensional power spectra we fit a model with asymmetric peak profiles (Basu and Antia 1999) specified by: P(k ,k ,ν) = exp(A0+(k−k0)A1+A2(kkx)2+A3kxk2ky)(S2+(1+Sx)2) x y x2+1 +eB1 + eB2, (1) k3 k4 where ν −ckp−U k −U k x x y y x= , (2) w +w (k−k ) 0 1 0 k2 = k2 + k2, k being the total wave number, and the 13 parame- x y ters A ,A ,A ,A ,c,p,U ,U ,w ,w ,S,B and B are determined by 0 1 2 3 x y 0 1 1 2 fitting the spectra using a maximum likelihood approach (Anderson, DuvallandJefferies1990).Here,k isthecentralvalueofkinthefitting 0 interval and exp(A ) is the mean power in the ring. The coefficient A 0 1 accounts for the variation in power with k in the fitting interval, while A and A terms account for the variation of power along the ring. 2 3 The term ckp is the mean frequency, while U k and U k represent x x y y the shift in frequency due to large scale flows and the fitted values of U and U give the average flow velocity over the region covered by the x y power spectrum and the depth range where the corresponding mode is trapped. The mean half-width is given by w , while w takes care 0 1 of the variation in half-width with k in the fitting interval. The terms involving B ,B define the background power, which is assumed to be 1 2 of the same form as that used by Patro´n et al. (1997). S is a parameter thatcontrols theasymmetry,andtheformofasymmetryis thesameas that prescribed by Nigam and Kosovichev (1998). The details of how the fits are obtained can be found in Basu et al. (1999) and Basu and Antia (1999). The fitted U and U for each mode represents an average of the x y velocities in the x and y directions over the entire region in horizontal extent and over the vertical region where the mode is trapped. We can invert the fitted U (or U ) for a set of modes to infer the variation x y in horizontal flow velocity u (or u ) with depth. We use the Regular- x y ized Least Squares (RLS) as well as the Optimally Localized Averages (OLA) techniques for inversion as outlined by Basu et al. (1999). The results obtained by these two independent inversion techniques are compared to test the reliability of inversion results. paper.tex; 5/02/2008; 5:26; p.3 4 S.BasuandH.M.Antia Figure 1. The power in ridges for n = 0,1,2 at different times is plotted against frequency, for region centered at the equator. The power is in arbitrary units. The continuous line is for Set 1, the dotted line for Set 2, the dashed line for Set 3 and dot-dashed line for Set 4. 3. Results We have fitted the four sets of summed spectra to calculate various mode parameters. In this work we are mainly concerned with the variation in flow velocities with time, but it is well known that other quantities also vary with time. The variation of mean frequency with time is well-known, butit is difficult to determine this small change us- ingringdiagramtechniqueastheimagescaleisknowntohavechanged with time to some extent. We do not find any significant variation in the width but the power in modes shows some temporal variation. Fig. 1 shows the power in n = 0,1,2 ridges in equatorial region as a function of frequency for each of the four sets of spectra. It is clear that power in Set 4, which covers a region of high activity is larger as compared to the other three sets which cover period of low activity. If this variation is real then the power in modes appears to increase with activity and the increase is quite significant at higher frequencies. Similartrendis seeninspectracentered atdifferentlatitudes. Thehigh frequency modes which penetrate into the near surface region are more likely to be influenced by surface activity and thus a larger increase in their amplitude indicates that the driving mechanism operates in the paper.tex; 5/02/2008; 5:26; p.4 SolarcyclevariationsoflargescaleflowsintheSun 5 Figure 2. The north-south antisymmetric part of the rotation velocity, i.e., [ux(N)−ux(S)]/2, plotted as a function of latitude at two different depthsfor all 4 sets of data. Theline-types are thesame as those in Fig. 1. near surface region, where the magnetic field is known to change with activity. 3.1. The rotation velocity The horizontal velocity u is known to be dominated by the rotation x rate, since the area under observation is tracked only at the surface rotation rate, and the solar rotation rate is known to vary with depth. Thusthemeasuredu arisesfromthedifferenceinrotationrateandthe x tracking rate. The ring diagram technique allows us to determine the north-southantisymmetriccomponentofrotationratewhichcannotbe inferred by the splittings of global modes. Fig. 2 shows the results at two different depths. There appears to be some temporal variation in thiscomponent,whichiscomparabletotheerrorestimates.Thereisno obvious pattern intemporal variation anditis notclear if theapparent differencebetweendifferentsetsisnotduetosomesystematicerrors.Of course, it is possible that the antisymmetric component of the rotation velocityvariesontimescalesoftheorderoftherotationperiod,whichis theseparationbetweenSets1and2.Suchvariationsintherotationrate at solar surface have been seen in Doppler measurements (Hathaway et al. 1996; Ulrich 1998). paper.tex; 5/02/2008; 5:26; p.5 6 S.BasuandH.M.Antia Figure 3. The smooth part of the solar rotation velocity (after subtraction of the rateatwhich theregions ontheSunweretracked)plottedasafunction oflatitude at two different depths for all 4 sets of data. The line-types are the same as those in Fig. 1. Following Kosovichev and Schou (1997), it is possible to decompose u into two components, a smooth part (expressed in terms of cosθ, x cos3θ and cos5θ, θ being the latitude) and the remaining part, which is identified with the zonal flows. Fig. 3 shows the smooth part of the rotation rate for all 4 data sets. This result automatically has the tracking-rate which is mean rotation velocity at solar surface, sub- tracted. There is some variation in the rotation velocity with time. However, the change between the low activity sets (1 to 3) may not be statistically significant. Results from global f-mode splittings show that there is a variation in the zonal flow pattern with time (Schou 1999) — the position of maximum and minimum zonal flow velocities drift equator-wards simi- lar to the torsional oscillation pattern seen on the solar surface.Similar resultshavebeenobtainedbyHowe,KommandHill(2000)andToomre et al. (2000) from inversion of frequency splittings. The ring diagram analysis provides a better depth resolution in near surface region as compared to the global modes and hence it can be employed to study thevariationwithdepthinzonalflowpattern.However,itisnotclearif thenon-smoothpartoftherotationvelocity isthesameasthetorsional oscillations, since the latter is the time variation in rotation velocity, and there could be some variation in smooth component which would paper.tex; 5/02/2008; 5:26; p.6 SolarcyclevariationsoflargescaleflowsintheSun 7 Figure 4. The zonal flow velocity plotted as a function of latitude at two different depths.The line-typesare thesame as those in Fig. 1. alsocontributetothetorsionaloscillations.Thisvariationisinfactseen fromourresultsinFig.3.Ideally,wecandetermineanaverage rotation rate over a long time period and subtract it from each measurement to get the time variation, which can possibly be identified with torsional oscillations. With only four measurements in time it is not possible to takeanymeaningfulaverageandhenceweattempttoidentifythezonal flows with the non-smooth component as has been done by Kosovichev and Schou (1997) and Schou (1999). It is known from earlier studies that the symmetric component of the zonal flow velocities agree well with the results from f-mode anal- ysis (Basu et al. 1999). Fig. 4 shows the symmetric component of the zonal flow velocities for all the data sets at two different depths. Note that there is some variation in the results with time. The change is particularly significant between sets 3 and 4. Solar activity is known to have changed quite rapidly between these two epochs. The latitudinal resolution of our study is not enough to clearly judge whether the changes between results of Sets 1, 2 and 3 are statistically significant, however,thereappearstobeashifttowardsequatorinthefirstmaxima in each hemisphere at a radial distance of 0.995R⊙. Results from Set 4 show the equatorial region to be rotating slower than the smoothed averagerotation rate,whileresultsfromthetheothersetsshowafaster rotation. This feature is also seen in the results obtained from f-mode splittings (Schou 1999). paper.tex; 5/02/2008; 5:26; p.7 8 S.BasuandH.M.Antia Figure 5. The zonal flow velocity plotted as a function of latitude at four different depths as marked in the figure. The panels on the left shows the result for Set 2 (low activity), while the panel on the right show the results for set 4 (high activity). In each panel, the points joined by the dashed lines are the results of an optimally localized averages (OLA) inversion. The continuous lines are the results ofaregularizedleastsquares(RLS)inversionwithdottedlinesshowingthe1σ error limits. Tostudythechangesinthezonalflowpatternwithdepthweconcen- trateonSets2and4.Fig.5showsthezonalflowvelocitiesforsets2and 4 at four different depths obtained using both RLS and OLA inversion techniques. The results obtained by the two inversion techniques agree with each other in all cases. We see that Set 4 in general has much larger flow velocities than Set 2. Furthermore, for the low activity set the direction of the flow appears to change sign as one goes deeper. Thisis notseen for the Set4. However, since theoverall flow velocity is small for Set 2, it is somewhat difficult to judge the significance of the change.FromtheresultsforSet4,itappearsthatthezonalflowpattern persists up to a depth of at least 0.02R⊙ during high activity period. Because of low amplitude it is not very clear if the zonal flow pattern penetrates to the depth of 0.02R⊙ during the low activity period too. There may be some change in pattern with depth and it is possible that the reversal in sign of zonal flow velocity at equator occurs after paper.tex; 5/02/2008; 5:26; p.8 SolarcyclevariationsoflargescaleflowsintheSun 9 Figure 6. The average zonal flow velocity at r = 0.995R⊙ plotted as a function of time (continuous line with crosses). The squares show the mean daily sunspot numberduringthesameperiodwhentheobservationsweremade,withthescaleon theright-hand axis. the solar minimum and starts at deeper layers, gradually advancing towards the surface, i.e., the reversal occurs earlier in deeper layers. Surfaceobservations appear to show that the zonal flow patterns is not well defined during the time around the solar maximum (Ulrich 1998). Thus it would be interesting to check what happens in deeper layers as we approach the maximum of cycle 23. Fig. 6 shows the magnitude of zonal flow velocity averaged over all latitudes included in this study, at a depth of 0.005R⊙, plotted together with the mean daily sunspot number. Although we have a very small sample, it appears that the mean flow velocity increases with increase in solar activity. Similar results can be obtained from study of global modes, for which data is available for larger duration, spanning a wider range of activity level. 3.2. The meridional flow The velocity component u is the meridional flow. Earlier results have y shown that the predominant flow pattern at the surface is from the equator to the two poles (Giles et al. 1997; Basu et al. 1999; Braun and Fan 1998). Fig. 7 shows the meridional flow velocity for the 4 sets at two different depths. It is clear that once again the Set 4 pattern is significantly different than the others, and in particular the ampli- tudes of high order component appears to be larger giving rise to some oscillations over the smooth pattern. paper.tex; 5/02/2008; 5:26; p.9 10 S.BasuandH.M.Antia Figure 7. The meridional flow velocities obtained from the 4 data sets plotted as a function oflatitudeat twodifferentdepthsmarkedin thefigure.Thelinetypesare thesame as those in Fig. 1. To take a more detailed look at the time variation, following Hath- away et al. (1996) we try to express the meridional component as uy(r,φ) = −Xai(r)Pi1(cos(φ)) (3) i whereφisthecolatitude, andP1(x)areassociated Legendrepolynomi- i als. Thefirstsix terms in this expansion are foundto besignificant and their amplitudes are shown in Fig. 8. The component a seems to show 6 themostsignificantchanges intheouterlayers wheretheinversions are most reliable. The amplitude of a6 around the depth of 0.005R⊙ ap- pearsto increase withsolar activity and may becorrelated to themean daily sunspot number. The amplitude of the dominant component, a 2 appears to have decreased slightly during the high activity period at depths around 0.005R⊙. The odd terms in the expansion, which rep- resent the north-south symmetric component of meridional flow, show much larger variation, particularly the first term a (r). The amplitude 1 a (r) of the first component appears to have reduced by about 5 m/s 1 in Set 4 as compared to Sets 1–3. In deeper layers the odd components of meridional flow appears to vary anomalously for Set 3. It is not clear if these variations in the odd components are due to instrumental effects arising from change in pointing errors, some systematic error in our analysis process, or whether they represent real changes in the paper.tex; 5/02/2008; 5:26; p.10