THE ASTROPHYSICAL JOURNAL 541,2001January1 2001

THE ASTROPHYSICAL JOURNAL, 546 :528È541, 2001 January 12001. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ON THE USE OF CORRELATIONS TO DETERMINE THE MOTIONS AND PROPERTIES OFMESOSCALE MAGNETIC FEATURES IN THE SOLAR PHOTOSPHERE

HERSCHEL B. SNODGRASS AND ADAM A. SMITH

Physics Department, Lewis and Clark College, 0615 Southwest Palatine Hill Road, Portland, OR 97219Received 1999 July 12 ; accepted 2000 June 9

ABSTRACTThe use of correlations to determine both intrinsic properties and collective motions of patterns is

investigated, and the results are applied to the study of the magnetic features in the solar photosphere.Simulations with artiÐcial data are used as a bridge between theory and practical correlation calcu-lations. It is shown that the correlation amplitude as a function of lag can be used to determine not onlypattern displacement, but also feature sizes and lifetimes. It is found that reliable results are obtainedonly when a normalized correlation function is employed, and then only when the signal-to-noise level isgreater than D 1.5. For weak correlations, we show that this ratio must be enhanced by averaging thecorrelation amplitudes, but when applied to the photospheric magnetic Ðeld patterns, this gives a resultdi†erent from that obtained by averaging the individual correlation results. We Ðnd this to be the rootof the di†erences between the magnetic rotation rates that have been reported and resolve this long-standing puzzle.

The correlations indicate the ubiquitous presence of di†erentially rotating magnetic features of twotypes : small-scale features that have lifetimes of D 1 day, and ““mesoscale ÏÏ features with lifetimes ofmany solar rotations. The latter are estimated to have diameters on the order of 100 Mm, and theirmotions relative to the ambient plasma are consistent with a random walk with di†usion constant D

m\

530 ^ 100 km2 s~1. Our value for agrees with that required in the model of Sheeley, Nash, & Wang,Dmbut these features are too large to have their random walks propelled by the supergranular convection.

Furthermore, analysis of their relative contributions to the background Ðeld implies they decay at a rateconsistent with a smaller di†usion constant km2 s~1. This agrees with the value determined inD

s^ 250

high-resolution studies, which suggests that the mesoscale features are aggregates of small-scale featuresundergoing random walks as well, like those observed in these studies.Subject headings : magnetic Ðelds È Sun: di†usion È Sun: rotation

1. INTRODUCTION

One of the several ways of studying the SunÏs rotationand meridional circulation is through correlations of suc-cessive maps of the photospheric magnetic Ðeld determinedfrom magnetograph observations. The features in these pat-terns are cross-sectional slices of Ñux tubes that extend todeeper regions. The individual Ðeld elements, which may beparts of larger structures, are a few hundred kilometers indiameter. These rapidly settle along the granular and super-granular cell boundaries where local horizontal convectivemotions are minimal, and appear to slip along these bound-aries as the convective cells evolve. Their mean motions aretherefore di†erent from that of the photospheric plasma atthe line formation depth.

Whereas the problems of interpreting photosphericDoppler shift measurements have been much discussed (e.g.,Beckers & CanÐeld 1976, Dravins, Lindegren, & Nordlung1981 ; Snodgrass 1992a ; Ulrich et al. 1988 ; Carter,Snodgrass, & Bryja 1992), the interpretation of correlationmeasurements involves di†erent challenges that havescarcely been addressed. In this paper, we shall not considerproblems of tracking individual features such as sunspots(e.g., Howard, Gilman, & Gilman 1984) or isolated Ðeldelements (e.g., Hagenaar et al. 1999), which is a laboriousprocess requiring judgments that are hard to quantify. Weshall focus instead on the use of correlations to determinemean, large-scale properties and motions of patterns, whereone does not follow any individual features but insteadtakes advantage of their large numbers and the randomness

of intrinsic changes. The features are required only topersist in recognizable patterns for ““ long enough ÏÏ times.

The mathematical foundations of correlation analysis arevarious theorems wherein the correlation functions areexpressed as integrals with Ðxed limits (e.g., Jenkins &Watts 1969 ; Bendat & Piersol 1971). These theoremsconnect the correlation amplitude, as a function of the lag,to the mean size, persistence, random and collectivemotions of the features being correlated, but idealize thedata as inÐnite or periodic continua. Properties of corre-lation amplitudes for real data have been discussed in thecontext of photographic methods in several Ph.D. theses(Noyes 1963 ; Simon 1963 ; Mosher 1977). In recent work,however, the calculations are done by computer for digi-tized data such as that taken in magnetograph obser-vations. The digitized data are binned into arrays of Ðnitesize, sums are used in place of the integrals, and the sumsmust extend only over the overlapping portions of thearrays, so that either the number of terms summed mustvary with the lag, or some form of ““ padding ÏÏ must beemployed. An investigation of validity of the theorems forsuch calculation algorithms has not been done.

The need for this investigation is evident. In the study ofmagnetic rotation, di†erent algorithms yield distinctly dif-ferent rates (e.g., Howard 1984 ; Snodgrass 1992a), andattempts to explain this lead to inconsistent conjecturesabout feature sizes and lifetimes (Snodgrass 1992b). In thispaper, we brieÑy review the various studies that have beendone, and point out disparities. Using numerical simula-

528

MESOSCALE MAGNETIC FEATURES 529

tions with artiÐcial data, we then investigate the algorithmsand address (1) the choice of correlation function, (2) therelation between decorrelation and pattern decay, (3) therelation between correlation peak width, features size, andrandom motions, and (4) the relation between the lags ofcorrelation peaks and the actual displacements of the fea-tures or patterns. We describe the use of correlation averag-ing to reduce noise and determine its e†ects on the results.With these foundations, we resolve the puzzle of the di†er-ing magnetic pattern rotation rates and draw several con-clusions about the mesoscale organization, persistence, andmotions of the SunÏs surface background Ðelds.

2. INFERENCES AND DISPARITIES IN PRIOR STUDIES

2.1. Rotation and Evolution of Photospheric MagneticField Patterns

The principal correlation studies of the SunÏs large-scalepatterns of motion have utilized the data collected in threelong-term synoptic magnetograph observation programs :the 46 m Tower Telescope program at Mount Wilson, theVacuum Telescope program at the National Solar Observa-tory at Kitt Peak, and the program at StanfordÏs WilcoxObservatory. A variety of choices of algorithm have beenmade on every aspect of the calculation, including the typeof correlation, the speciÐc correlation function, the pro-cedure used for locating the correlation peak, and the wayin which averagings are done.

Examination of an individual magnetogram or of a Car-rington map such as that shown in Figure 1, taken when theSun is active, shows plumes of alternating magnetic polaritythat arc poleward from the low-latitude activity centers. Athigh latitudes they become horizontally elongated and takeon the appearance of nearly azimuthally symmetric waveswhich lap at the poles. The corresponding neutral lines, seenin Ha as Ðlament channels, are called the polar crown.

Most of these plumes appear to rotate rigidly, as if theywere attached to the active regions. One would guess, there-fore, that the measured rotation of the solar magnetic Ðeldswould also be mostly rigid, and hence very di†erent fromthe di†erential rotation of the photospheric plasma. This

was noted by Bumba & Howard (1969), using data collectedearly in the Mount Wilson program. Soon thereafter, thetime series of Ðeld values along the neighborhood of thecentral meridian for each dayÏs observation was autocorrel-ated (Wilcox & Howard 1970 ; Wilcox et al. 1970 ; Schatten,Wilcox, & Ness 1972 ; StenÑo 1974, 1977), and the resultsconÐrmed this almost-rigid rotation of the Ðeld patterns.The same rigidity was found in the rotation of structures inthe corona (Timothy, Krieger, & Vaiana 1975 ; McIntosh,Willock, & Thompson 1976).

This result was consistent with the then-prevalent ideathat the di†erential rotation was produced within the con-vection zone ; it suggested that the Ðeld is rooted at a depthwhere solar rotation is more rigid than at the surface. Buthelioseismology has now shown that the di†erential rota-tion extends with little change to the base of the convectionzone, so this explanation is unlikely. It must have seemedpeculiar even then, however, since (1) sunspots, while con-Ðned to low latitudes, showed di†erential rotation (Newton& Nunn 1951) ; (2) high-resolution studies of high-latitudeÐelds showed only small elements that did not appear to bemoving rapidly relative to the photospheric plasma(Howard 1978) ; and (3) the idea of high-latitude Ðeld struc-tures plowing through the solar plasma at up to D 250 ms~1 as required by this di†erence in rotation rates deÐes thelaws of physics.

The mystery was brought to focus when a study of thecross-correlations of magnetograms spaced by time inter-vals (““ lag times ÏÏ) of 1È4 days (Snodgrass 1983, hereafterPaper I) found the rotation of the magnetic features to benot rigid, but instead to follow essentially the same di†eren-tial rotation proÐle as the photospheric plasma, thoughD2% faster at all latitudes, and to agree precisely with theNewton & Nunn (1951) sunspot proÐle in sunspot latitudes.Soon thereafter, Sheeley, Nash, & Wang (1987, hereafterSNW) did a cross-correlation of successive Carringtonmaps (maps of the ““ whole ÏÏ Sun made by overlapping aver-ages of the daily magnetograms), but they obtained themore rigid rotation proÐles. StenÑo (1989, 1990) thenreworked his autocorrelation study and conÐrmed his

FIG. 1.ÈCarrington map of the solar magnetic Ðeld during the active phase of the cycle. This map is for Carrington rotation 1816 (mid-1989), and the Ðeldwas measured in the Fe I spectral line at j5250.2.

530 SNODGRASS & SMITH Vol. 546

earlier rigid rotation results. He noted that the peak widthin the correlation amplitude was, at least at high latitudes,far broader than that found in Paper I, suggesting that thefeatures that dominated his correlations were larger.Finally, however, Komm, Howard, & Harvey (1993) evenedthe score ; by doing high-resolution cross-correlations of theKitt Peak magnetograms spaced at 1 day intervals, theyconÐrmed the di†erential rotation found in Paper I.

Figure 2 shows these various rotation proÐles. How thesedi†erent correlation calculations pick up the two di†erentproÐles and what the two proÐles mean have been subjectsof debate (StenÑo 1990, 1992 ; Snodgrass 1992a ; Wang &Sheeley 1994 ; Sheeley & Wang 1994), which we hoperesolve in this paper.

2.2. T he Di†usive Flux Dispersal ModelIn the correlation studies where di†erential rotation was

found, the lag times were short and the peaks in the corre-lation amplitude were narrow and fell o† rapidly with lagtime. The more rigid rotation proÐle was found in long lagtime studies, where the peaks were found to be broad andlong lasting. Though logic did not dictate it, this suggestedthat the short lag-time studies were dominated by smallfeatures in the Ðeld, which died away or were dispersed in afew days, so that less common, but larger and longer livedfeatures like the plumes seen in Figure 1 could dominate thelong lag time correlations.

By incorporating this suggestion into a model proposedoriginally by Leighton (1964), SNW were able to accountfor the simultaneous presence of patterns of rigid and di†er-ential rotation. In this model, all of the solar background

FIG. 2.ÈSolar rotation using various indicators. All proÐles except thatlabeled Doppler shifts are determined by cross-correlation. The di†eringmagnetic rotation proÐles are labeled quiet magnetic for the result thatparallels the Doppler proÐles, and active magnetic for the more rigid rota-tion.

Ðeld is produced through the di†usive decay of the active-region Ðelds. The small features are active-region fragmentsthat have broken o†. Relative to the di†erential rotationand poleward meridional Ñow of the ambient plasma, thesefeatures move with a random walk propelled by the super-granular convection cells. Their concentration into plumesis analogous to the rising of smoke in columns from smoke-stacks in a wind : as the individual particles both rise anddrift horizontally, the columns as a whole remain attachedto the stacks, and stationary. On the Sun, the plumes aresimilarly attached to active-region sources. Within eachlatitude band poleward of the active region, the concentra-tion rotates at a ““ phase ÏÏ velocity which is faster than therotation of the individual fragments since the fragments arecontinually arriving at the west end from lower latitudesand leaving at the east end for higher latitudes. As higherlatitude bands are further removed from the sources, theÐeld elements take longer to reach them and thus theplumes slope back away from the direction of the solarrotation. Since the higher latitude bands also rotate pro-gressively more slowly, the plumes also curve backward, i.e.,eastward. The crossing of such concentrations seen inneutral-line stack plots (McIntosh et al. 1991) occursbecause the active region sources are at di†ering latitudes.

This model overcomes the objection about magneticÐelds plowing through the solar plasma at high rates ofspeed, for the Ðeld elements simply move with the variousmotions of the plasma. The transport of the Ðeld over largedistances is discussed in a Ph.D. thesis by Mosher (1977),who concluded, on the basis of an order-of-magnitude argu-ment about magnetic tension, that the supergranular con-vection is a viable mechanism. More recently, however,Wilson, McIntosh, & Snodgrass (1990) argue from con-siderations of both tension and buoyancy that this shouldnot be possible, so this basic matter of the physics of Ñuxtubes is not yet resolved.

A problem for the Leighton-SNW model is that the di†u-sion constant is too large. Taking the Doppler-measuredbulk poleward meridional Ñow of m s~1, they Ðndv

mB 10

D^ 600 km2 s~1 is required in order that the model give areasonable simulation of the large-scale Ðeld patterns seenon the Sun and the polar Ðeld reversals of cycle 21. This is,however, D 2 times larger than the value both anticipatedin simple calculations and deduced from actual Ðne-scaleobservations of the motions of Ñux elements (Mosher 1977 ;Schrijver & Martin 1990 ; Hagenaar et al. 1999). It is alsoD 3 times larger than the value found by Komm et al.(1995) from Gaussian Ðts to the correlation amplitudes fortwo-dimensional, 1 day lag time cross-correlations of high-resolution magnetograms taken at Kitt Peak.

In the past several years, some interesting theoreticalwork has been done (e.g., Lawrence & Schrijver 1993),showing that supergranular di†usion of the Ðeld elementsought to be regarded as fractal, of dimension less than 2,because it is constrained to occur not in all directions butonly along the shifting boundaries of the supergranules,where the Ñows can act as ““ one-way ÏÏ streets for the Ðeldelements. In this paper, however, the data are too noisy tomerit attention to this reÐnement, and we shall to couch ourdiscussion in terms of the simpler two-dimensional picture.

2.3. Problem of Feature L ifetimesThe di†usive decay time of features of longitudinal wave-

number k is on the order of (Dk2)~1. With D\ 600 km2

No. 1, 2001 MESOSCALE MAGNETIC FEATURES 531

s~1, large-scale features such as the rigidly rotating plumesor km) should have lifetimes of(*/Z 50¡, L Z 5 ] 105

several solar rotations, and the 10 times smaller featuresthought to dominate the short lag-time correlations wouldlast only a few days. These decorrelation times agreed wellwith the times found in Paper I and in Komm et al. (1993).However, as StenÑo (1994, private communication) pointedout, regardless of the value of D, di†usion cannot producesmall-scale clumpings of features, and therefore no smallfeatures should be found at any appreciable distance froman active region, which runs up against the fact that they arefound all over the Sun.

In response to StenÑoÏs objection, Wang & Sheeley (1994,hereafter WS) ; see also Sheeley & Wang 1994) pointed outthat the di†usion, if produced by the supergranular convec-tion cells, must actually be treated, as originally suggestedby Leighton, as a random walk of indestructiblesupergranule-sized ““ features ÏÏ for which steps of length Lare taken at time intervals q, where L and q are, respectively,the radius and lifetime of a supergranule. Such a randomwalk does not take apart the features and looks like a di†u-sion with constant at scales large comparedD\ 14L2q~1with L and q (which gives D in the range 200È600 km2 s~1).

WS did a series of simulations of this random walk, whichproduced maps that looked more like actual magnetograms(see Fig. 1) than those from their earlier di†usion studiesbecause they exhibited both the ubiquitous clumpiness andthe plumes. Upon cross-correlating these maps, they foundthat for short lag times the averaged (see ° 3.2 below) corre-lation amplitude had a sharp spike corresponding to thedi†erentially rotating supergranule-sized clumps, which wasperched atop a much broader peak corresponding to therigid rotation of the plumes. For long lag time day)(Z 15

correlations, they found that the sharp peak disappearednot because the clumps had disintegrated but because theyhad become so rearranged that their pattern was unrecog-nizable. This was consistent with the discussion in Mosher(1977 ; cf. eq. [4.47]), where it was shown that, owing to therandom walk, the supergranule-sized feature correlationamplitude should be reduced to P^ 0.3q/t, or P^ 0.01after a full solar rotation.

In the light of these discussions, it seemed evident that ifcorrelations of the Paper I type were done with long lagtimes, the peak corresponding to di†erentially rotating Ðeldstructures should not be present in the amplitude. But onattempting to verify this by doing cross-correlations of indi-vidual magnetograms at lag times *T of 1È3 full solar rota-tions, as opposed to 1È4 days, (Snodgrass 1992b ; Snodgrasset al. 1993), it was found that the di†erential-rotation peakwas still dominant, and an order of magnitude larger thanwould be estimated on the basis of the WS supergranule-scale random walk. The exact rate, curiously, was foundalso to be D 0.5% faster at all latitudes than that found inPaper I.

Sample plots of the correlation amplitude for a short lagtime (3 days) and a long lag time (30 days) are seen in Figure3. The top panels correspond to a period around solarminimum, and the bottom panels correspond to a similarperiod around solar maximum. The smooth and well-re-solved contours in these maps are revealed by correlationaveraging over 2 years to reduce the noise (this procedurewill be discussed below in ° 3.2). The normal di†erentialrotation is clearly seen in the ridge lines, and is particularlyclear when it is more spread out in the 30 day case.

Figure 4 shows the peak amplitudes, i.e., the amplitudesalong the ridge lines like those seen in Figure 3, plotted

FIG. 3.ÈTwo-year time averaged cross-correlation amplitudes for magnetograms separated in time by 3 days (left panels) and by 30 days (right panels).The top panels are for a quiet Sun (1986È1987), and the bottom panels are for an active Sun (1989È1991).


FIG. 4.ÈCycle 21 averaged peak cross-correlation amplitude as a func-tion of latitude for magnetograms separated by 3 days (0 rotation), by 30days (1 rotation), and by 55È59 days (2 rotations).

against latitude. In this plot, the correlations have beenaveraged over all of cycle 21. Since the Pearson cross-correlation function (see Appendix A) is normalized, theamplitude for autocorrelation (*T \ 0) is unity ; thusFigure 4 shows that the decorrelation is rapid over the Ðrstfew days, particularly at high latitudes. This was noted inpaper I and had fueled the speculation that the features areshort-lived. Figure 4 further suggests, however, that, fea-tures surviving the Ðrst few days had a much slower decayrate (see ° 5.2 below).

The persistence of a ““ smaller ÏÏ scale, di†erentially rotat-ing feature amplitude at high latitude was actually alreadyoccasionally seen in the autocorrelations, as shown inFigure 5, taken from StenÑo (1989). The autocorrelationamplitude at low latitude ([13¡) contains a ““ Ðnestructure ÏÏ suggesting the presence of several active regions,and the narrowness of the peaks attests to the small featuresizes. At the high latitude ([75¡), the correlation amplitudeexhibits (above a large noise component) a very broad peakrotating and decaying at close to the same rate as the sharplow-latitude peaks. This dominates the correlation out toabout 4 Carrington rotation periods, and its width andshape suggest an E-W magnetic dipole Ðeld rotating atapproximately the same rate as the low-latitude Ðne struc-ture. It is too broad to be picked up in the cross-correlations

FIG. 5.ÈAutocorrelation amplitude vs. lag time. L eft : 13¡S ; right : 75¡S.From StenÑo (1989), Fig. 1. Solid vertical lines indicate the peak positionsto determine the rotation periods, and dashed vertical lines indicate therotation periods determined by the cross-correlations in Paper I.

of individual magnetograms. Note, however, the verticaldotted lines in Figure 5 which point out a second peakÈalittle peak which rotates at the Paper I di†erential rotationmagnetic rate. At Ðrst this little peak is seen as a small bumpon the larger ““ dipole ÏÏ amplitude, but it does not decay asrapidly as the latter, and after Ðve ““ dipole ÏÏ rotations itbecomes the dominant peak.

Thus the actual data show that there are features whichare smaller than the plumes and yet persist for very longtimes ; whether they persist for long enough times to migratefrom the active-region latitudes to the poles will be dis-cussed in ° 5. That the decorrelation of these features is lessrapid than predicted in the WS simulation suggests, in viewof MosherÏs (1977) discussion, that they are larger thansupergranules. We will Ðnd other evidence for this in ° 5.

2.4. Unresolved Question of the Di†ering RatesThe question of how the di†ering correlation schemes

pick up the di†erent rotation rates is therefore not fullyresolved in the WS simulation. As we have noted, when thePaper I type correlations are done for even very long lagtimes, the di†erential rotation proÐle is still dominant, andthere is no evidence in Figure 3, for either the short or longlag-time amplitudes, of a sharp spike perched atop a broadpeak. On the other hand, in the SNW correlations and inmost of the autocorrelation studies, it is the broad ““ plume ÏÏpeak that is dominant. To understand this, we must lookmore carefully at the correlation algorithms.

As has been pointed out by SNW and others, bothresolution and time delay are important factors. The Ðelditself is known to consist mainly of subarcsec size features,and unipolar regions in the large-sale Ðeld are regions inwhich there is a preponderance of features of one polarity. Ifthe resolution is high enough to distinguish these individualfeatures, and if the time interval is short enough that theirrandom motions and changes can be ignored, the corre-lations would not register the plume rate. If, on the otherhand, either the resolution is very low, or the time intervalso long that individual features are thoroughly mixed andspread out, only the overall concentration would be pickedup, and the plume rate alone would be found. For interme-diate resolutions and times, the amplitude should containpeaks corresponding to both rates, as in the WS simulation.

In using the correlation amplitude to determine the meanmotions of features, one Ðrst identiÐes a relevant peak in theamplitude and then employs a curve-Ðtting routine tolocate its center. In Paper I, the peak is identiÐed simply byÐnding the highest point in the amplitude, and a three-pointparabolic Ðt is employed to Ðnd its center. In the WS simu-lations, where two peaks are clearly seen, they correctlypoint out that the three-point parabolic Ðt used to deter-mine the center of the sharp, small-feature peak is notappropriate for determining the center of the much broader““ plume ÏÏ peak upon which it is perched. But examination ofFigure 3 shows little or no evidence of the broad peak seenin the WS simulation ; hence their curve-Ðtting argument isnot relevant. The reason the broad peak is missing is thatthe ( D 90¡-wide) window available on individual magneto-grams is too narrow; the regions of unipolar Ðeld that giverise to the broad peak are so broad that they are nearlyconstant or linearly varying over this window, and thus notregistered in the correlation amplitude.

This suggests that the di†erence in correlation-determined rotation rates might simply be due to the width


of the data window. In a preliminary study to investigatethis, we performed Paper I type correlations on the suc-cessive Carrington maps used by SNW. Whereas we nowfound both a broad peak and the sharp peak in the resultingamplitudes, we found that the sharp peak from the di†eren-tially rotating features was still dominant even for a timeseparation of two full Carrington rotations, and the deter-mined rotation rate was the same as that obtained by corre-lating individual magnetograms at long lag times. We thusconcluded that while the width of the data window is rele-vant, it is not the root of the discrepancy.

3. CORRELATION ALGORITHM

The correlation algorithm includes not only the corre-lation function employed, but also the manner in which thedata are interpolated into bins, the number and size of thesebins, the procedures for locating and determining the lag orlags for the maximum or maxima in the correlation ampli-tude, and Ðnally, the procedures for averaging.

In many studies (speciÐcally SNW and WS), the functionthat has been used to determine rotational displacement is

deÐned in equation (A1) in the Appendix A, com-Cfg

(L ),monly called the cross-covariance. This function, however, isa biased estimator when the Ðelds f and g are not periodicon the Ðnite domain over which the sums are carried out.We note also, in passing, that the calculation is then fre-quently done using a fast Fourier transform, which is stan-dard in many o†-the-shelf data analysis packages currentlyavailable. The Ðniteness and variation of the overlapdomain with the lag L are handled in a number of ways,such as padding with zeros, or forcing the data to be period-ic. We shall not discuss this approach since in our applica-tions it does not facilitate the calculation, but introducesextra steps and systematic error which add to the numericalnoise.

Correlation functions are discussed in the Appendix A,where we show that the Pearson cross-correlation, P

fg(L )

(eq. [A3]) is the most appropriate for our purpose. Thisamplitude varies from ]1 for perfect correlation to [1 forperfect anticorrelation. It is easy to compute ; incorporatingthe lag-dependent domain leaves us only with an insigniÐ-cant bias in the correlation amplitude that arises from thediscreteness in the domain (i.e., the binning of the data). Thecomplexity of the Pearson formula, however, imposes prob-lems for formal mathematical proofs of some of the proper-ties we wish to verify. We get around this difficulty with theuse of simulations in ° 4.

3.1. Autocorrelations and Cross-CorrelationsAs we have noted, both autocorrelations and cross-

correlations have been used to determine the rotation of thesolar magnetic Ðeld. Autocorrelations reveal structures withpersistent periodicity in longitude, and cross-correlationsgive mean displacement over a range of longitudes. Thesame mathematical methods are applicable to both.

For autocorrelations, f and g are the same function. In atime-series autocorrelation, the time series for each latitudeis the set of Ðeld values measured over a longitude windowin a sequence of observations. Between observations, theSun rotates ; hence, this time series is a sampling of the Ðeldat both successively later times and longitudes, the latterstepped in intervals equal to the product of rotation ratewith time between observations. This longitude samplinginterval determines an azimuthal ““ spatial resolution ÏÏ

which limits both the accuracy to which feature sizes can bemeasured and the resolution of longitudinal dependence.For one observation per day, which is the frequency for thedata represented in Figure 5, the limit is D13¡ at the lowlatitude and D11¡ at the high latitude. The narrowest peaksseen on Figure 5 are approximately at this limit.

The autocorrelation amplitude should, if the Ðeld sur-vives a few solar rotations, show periodicity with decreasingamplitude at integer multiples of the rotation period(s).Examples of this are seen in Figure 5. The rotation periodsare determined by measuring the lag time between suc-cessive principal maxima.

In using cross-correlations to study rotation, the domainis solar longitude /, which is measured relative to a speci-Ðed meridian. The quantity L is thus lag in longitude, andthe data streams f and g are the Ðeld arrays from separatemagnetograph observations, or from separate Carringtonmaps, spaced at some chosen time interval *T . If *T \ 0,the correlation is called a ““ spatial autocorrelation,ÏÏ whichis useful for ascertaining feature size.

The spatial resolution in a cross-correlation is the bin sizeof the correlated arrays, and this limits the accuracy offeature size measurements. The resolution of longitudinalstructure, on the other hand, is limited to the width of thedomain over which the calculation is performed. The veloc-ity is determined as the lag of the selected peak in the corre-lation amplitude divided by *T . This determination isgenerally more accurate than that from the autocorrela-tions, and one is able to determine changes in rate withtime.

3.2. Averaging ProceduresConsider the determination, by the correlation of features

that decay rapidly, of a displacement that is expected to besteady in time. As we shall see in the simulations of ° 4,correlation of individual pairs of maps of these features willcontain a high level of noise which may totally obscure thesignal peak that represents the displacement in question.This can be remedied by averaging the correlation ampli-tudes from a succession of map pairs, and this method,correlation averaging, was employed in Paper I, and also byKomm et al. (1993). For this to work, the time separation*T between each correlated map pair must be exactly thesame. For data taken daily, but at di†ering times, this canbe accomplished, as shown in Paper I, by adjustment of thelongitude grid into which the data are binned. This adjust-ment, which requires an input estimate of the rotation rate,may be signiÐcant for short lag-time correlations but hasnegligible impact for lag times of a solar rotation or more.

It would seem that a viable or even preferable alternativeto this correlation averaging would be lag averaging,wherein the highest peak in each individual map-pair corre-lation is Ðtted to determine its lag, and the peak lags arethen averaged. It is found, however, that for Mount Wilsonmagnetograms at separations days, the location of*T Z 3the maximum begins to be random and sensitive to the datawindow width, and so this approach fails to give consistentresults. This will be quantiÐed in ° 4.2.

On the other hand, correlations of successive Carringtonmaps give relatively consistent results without correlationaveraging. The reason may be in part as SNW suggest : thatthe full 360¡ longitude range now available allows the corre-lations to pick up very large features, such as the dipole-likepeak seen in Figure 5, which are not registered in corre-


lations of the D90¡ range available on individual magneto-grams, and the result, as noted above, is a di†erent rotationproÐle. Further discussion of this will be given in ° 4.3 andin ° 5.1, where we show how the result that is obtaineddepends on the averaging procedure.

4. SIMULATIONS

To investigate correlations of real data, which employÐnite sums with limits that vary, we do numerical experi-ments with artiÐcial data, varying one factor at a time todetermine its e†ect. Both the normalized correlation C andthe Pearson correlation P described in the Appendix A aresound choices for the correlation function, but our numeri-cal studies show that P, which has been employed in PaperI and most other works, has the slight advantage that thepeak is more distinct, and easier to resolve above therandom noise. We conÐne our further discussion to theproperties of this function.

4.1. ArtiÐcial DataOur artiÐcial data are tailored to resemble Carrington

maps of the magnetograph data obtained at Mount Wilson,and the procedures are the same as those in Paper I. Thereal-data bins are square and 4¡ on a side, and the latituderange is between ^72¡. Accordingly, our artiÐcial data arebinned into 36] 90 arrays of 4¡ ] 4¡ bins. As our goal issimply to study the properties of the correlation amplitude,we treat the 36 strips as a statistical ensemble of 36 separatenumerical experiments.

Cross-correlation involves a pair of maps separated intime, and during this time the Ðeld in the Ðrst map hasevolved both through large-scale advection and throughrandom motions and changes including the decay, appear-ance, and disappearance of features. In the initial array F ofour artiÐcial data, each bin is assigned a random number inthe range ^100. To simulate the random changes thatoccur during the time *T between observations, the secondarray G is constructed by creating another random array Hand adding it and the original array together in variousproportions : G\ aF] (1[ a)H with 0 \ a \ 1. The factora characterizes ““ feature survival,ÏÏ (1[ a) gives the extent towhich the features in the original array have faded into thebackground noise, and the signal-to-noise ratio is S/N \ a/(1[ a). The correlation of the two arrays F and G providesus with a measure of the relationship between decay oforiginal features and correlation amplitude.

From these original arrays we create arrays having largerfeature size by smoothing over varying numbers of bins. Bycorrelating these we see how feature size is reÑected in thewidth of the correlation peak. To simulate motion, we dis-place the numbers in the bins (adding new random numbersinto those that have been vacated) in array F, before pro-portionally adding it to H. We then compare the lag ofL

Mthe Ðtted maximum correlation-peak in with thePFG

(L )input displacement.

For the range of 90 longitude bins, we allow the totalrange of lags to be 65. The location and shape of the corre-lation peak is, in each case, determined through the use of athree-point parabolic Ðt. To study correlation averaging, weconstruct a series of 100 di†erent random 36 ] 90 arraypairs F and H and treat them identically as describedabove, so that varying numbers N of correlation amplitudescan be averaged before the peaks are located. While theexact numerical results that we describe in the followingtwo sections depend somewhat on the actual number ofbins and lag range we have employed, the general conclu-sions are relevant to all such correlation calculations.

4.2. Inference of Displacement, L ifetime, and SizeOf particular interest are the minimum requirements on

the survival factor a and the number of map pairs N in thecorrelation average. The Ðrst thing we discover in our simu-lations is that a reliable result cannot be secured unless thesignal-to-noise level S/N is at least 1.5, corresponding, withno correlation averaging (N \ 1), to a º 0.6. When corre-lation averaging is done (N [ 1), the noise decreases as theinverse square root of N. This is illustrated in Figure 6a,where we plot the minimum value of a for which theamin,lag of the maximum correlation peak matches the actualL

Mpattern displacement, against N : the solid curve is given by0.6N~1@2.

This demonstrates the importance of correlationaveragingÈwith it we can explore situations where thecorrelation is very weak. The boundary is remarkablysharp : for each choice of the number of map pairs N in thecorrelation average, the lag of the correlation peak matchesthe actual input displacement to better than 0.1% times(feature size measured in bin widths) when buta º amin,develops huge errors and quickly becomes random when

In the lag-averaging procedure, on the other hand,a \amin.the average converges quickly for very slowly fora Z 0.6,and fails to converge at all for (i.e.,0.6Z a [ 0.5, a [ 0.5

FIG. 6.ÈPearson correlations of 90 bin arrays of artiÐcial data. (a) Minimum value of feature survival amplitude a required for pattern displacementaminto be determined from the lag of the maximum peak in the averaged correlation amplitude, as function of number N of map pairs in average. (b)Decorrelation curve : averaged peak height as a function of a for various values of N. (c) Peak width at half-maximum, as function of feature length (bothmeasured in bins) for a [amin .


S/N ^ 1) where the result becomes dependent on thenumber of bins in the correlation sum.

In Figure 6b we show the relation between the height ofthe Ðtted correlation amplitude peak and the a coeffi-P

Mcient. The random errors for points on the smooth decorrel-ation curve are too small to show on the graph, but forpoints along the break-away portions, they exceed itself.P

MNote that the decorrelation follows a smooth curve provid-ed that where is determined in terms of Na º amin, aminfrom Figure 6a. Alternatively, for each value of a, there is aminimum number of terms needed in the correlationNminaverage before the smooth curve for is obtained. FigureP

M6b shows that averaging over more than this minimumnumber does not further a†ect the result, i.e., that isP

M(a)

independent of N for N Z Nmin.One sees in Figure 6b that the peak height remainsPMnear unity until the features have decayed by about 40%

(i.e., a \ 0.6), and then it drops steeply and nearly linearly tozero at a \ 0, provided that N is kept high enough. Thus for

we Ðnd that and from this, assumingPM

[ 0.8, a B 0.75PM

,a single exponential decay rate, the feature lifetime for

can be estimated asPM

\ 0.8 q^ *T /(0.3 [ ln PM

).In Figure 6c, we see that the width of the peak W (P

M),

which we deÐne as the width at half-maximum of the parab-ola with which the correlation peak is Ðtted, is nearly pro-portional to the feature size, here measured in bins. It isagain found that this holds for all in our artiÐciala [amindata, but there is cross-talk between and whenP

MW (P

M)

the feature size approaches the width of the data window orwhen the pattern displacement brings the peak near itsedge. Hence these conditions must be avoided.

Analytically, one can easily see that the width of thecorrelation peak depends on the featureÏs shape : for sine-wave features and rectangular (square-topped) features, thewidth of the correlation peak is the same as the featurewidth, but for Gaussian shaped features, the correlationfunction C would give it as wider by a factor of TheJ2.extraction of the means in the Pearson correlation P,however, e†ectively narrows the peak-width measurementas noted in the Appendix A. This is because half-maximumis measured from zero. The near equality between peakwidth and feature width, hence the decreased sensitivity tofeature shape, seen in Figure 6, is characteristic of thePearson correlation function.

Finally, consider a correlation in which the features in thelater map have changed size and have undergone somerandom motion. Both these widen the peak, and the widthrelationship shown in Figure 6c can no longer tell us thefeature size. If we suppose that both the enlargement andmotions are random processes, then if a and b are the meanfeature widths in the two maps and X is the mean distancethe features have moved, for the normalized correlationfunction C the peak width will be (a2] b2] X2)1@2 if thefeatures are Gaussian (see eq. [2] below). If, however thefeatures are rectangular and/or the Pearson correlation P isused, we get closer to (b2] X2)1@2, assuming b [ a.

4.3. L ag Averaging versus Correlation AveragingIn the preceding section we have seen that correlation

averaging is a critically important tool since it enables thestudy of correlations which are present but dwarfed byrandom noise. We have found that it does not have afurther systematic a†ect on the amplitude, width, or dis-placement of the correlation peak, once the number of

correlations averaged is sufficient to obtain a signal-to-noise level Z 1.5.

The correlation-averaging and lag-averaging methods donot give precisely the same result even when the individualmap-pair correlations are strong, however. First, as wasshown in Paper I, if the maximum correlation value alwaysoccurs at the same integral bin-value of the lag, as it willwhen the correlations are strong, correlation averaging fol-lowed by a three-point parabolic Ðt is equivalent to weigh-ting the lag average by the ““ sharpness ÏÏ of the peaks.

A second potential source for di†erences in the obtainedresults, however, is far more signiÐcant : Thus far we haveconsidered only simulations where the pattern moves as awhole with a Ðxed rate, giving rise to a single peak in thecorrelation amplitude. The discussion in ° 2.1 shows,however, that on the Sun there can be two or more collec-tive motions with di†erent rates, each giving rise to a peakin the correlation amplitude. Our algorithms only searchfor the highest peak ; if there are two rates, the one whichgives rise to the highest peak can depend not only on whichtype of features dominates, the resolution, and the datawindow, but also on the choice of averaging method. Forconsider the case of two collective motions, both givingstrong correlations. Let one have a variable presence andrate of motion, and the other have a Ðxed rate which isalways less than the variable rate. Individual map-paircross-correlation amplitudes will then have two peaksÈonewhich has a variable height and a variable lagP1 L 1(P1),and another which has height always at a Ðxed lagP2 L 2.The lag-averaging method will give a mean displacement

whenever in one or more of the individ-L LA[ L 2 P1[ P2ual amplitudes. The correlation average displacement L CA,on the other hand, will equal if either the average peakL 2height of the variable peak is less than that of the Ðxed peak,or if the variability in the rate washes the contributions of

out.P1To simulate the e†ect of the variable rate we copied ourrandom-data maps F to two maps, and and weF1, F2,smoothed to give it larger features. The small-featureF2arrays were given a Ðxed displacement and the large-F1 x1,feature arrays were given larger but somewhat variableF2,displacements The G maps were now made as G\x2. Choosing slightly12[a1F1 ] a2F2] (2 [ a1[ a2)H]. a2greater than individual correlations of F and G yielded,a1,as expected, two peaks with usually dominant. It wasP

F2remarkably easy to the make the small feature-rate peakdominant in the correlation average, however, owing bothto the above-noted ““ sharpness ÏÏ weighting and to thewashing-out of the contributions, because of their di†er-P2ing locations.

5. ANALYSIS OF SOLAR FIELDS

We now apply the conclusions reached using simulationsto the analysis of correlations of the actual solar data. Weconsider both the Mount Wilson magnetograph data in34 ] 34 coarse-array form, and Carrington maps of thesedata.

5.1. Rigid versus Di†erential RotationThe most prominent background-Ðeld features seen on

coarse-scale magnetograms are the plumes, which form uni-polar arcs extending poleward from activity complexes.Many of these appear to rotate rigidly. Less evident, butclearly identiÐable in auto- and cross-correlations of these


magnetograms are features that are unattached, scatteredall over the Sun, and follow the di†erential rotation of thesolar plasma. The highest peak in a cross-correlation ampli-tude can be from either of these feature types, althoughsince the active regions come and go, the plume contribu-tions are more variable. An algorithm that determines therotation rate from the displacement of the highest peak willÐnd the plume rotation (PR) rate in some cases and thedi†erential rotation (DR) rate in others, and therefore, sincefor latitudes the PR rate is always the faster, theZ 35¡average of the rates determined separately from each indi-vidual correlation will also be faster than the DR rate,becoming more like a rigid rotation when the fraction ofplume-dominated correlations is greater.

When the correlation amplitudes themselves are aver-aged, however, the plume contribution becomes washed outbecause it varies in position and strength and is sometimesabsent altogether. Our investigation in ° 4.3 suggests thateven when the DR peak is weaker in the majority of theindividual correlation amplitudes, it better survives theaveraging process since it is always present and alwaysoccurs at the same location. Hence, the correlation averag-ing method almost invariably yields the DR rate.

We conÐrmed this for the series of Carrington maps, cor-related as described in ° 2.4. Without correlation averaging,we obtained both PR and DR proÐles. Examination of thecorrelation amplitudes showed the PR peak to be broaderthan the DR peak. When prominent plumes were presenton the maps, it was often stronger, but at other times it wasweaker or even absent. Averages over the lags of the Ðttedmaximal peaks yielded rates greater than the DR rate, andcloser to the PR rate when activity was high. We thusobtained the intermediate, somewhat irregular proÐlesfound by SNW, seen in Figure 2 above. On the other hand,except at polar latitudes, averaging the correlations overonly three map pairs was usually sufficient to yield the DRrate found in Paper I.

The 2 year averaged correlation amplitudes seen inFigure 3 are all from the cross-correlations of individualmagnetograms. Each of these yields a rotation proÐle that isnearly identical to the di†erential rotation proÐle (Fig. 2)found in Paper I ; however, the rate is D 0.5% faster for the30-day lag times. The ““ horns ÏÏ seen in both lower (active-Sun) panels are the vestigial contributions of the rigidlyrotating plume amplitudes.

Thus it is correlation averaging, which was done in PaperI simply to scrape away the noise to better resolve the peakin the amplitude, that tips the balance toward the DRproÐle. Only at the extreme polar latitudes does the plumerate peak occasionally dominate this averaged amplitude,and here it is always very broad, suggesting, as in Figure 5,the rotation of an o†-axis dipole Ðeld.

5.2. Spectrum of Sizes of the Di†erentially Rotating FeaturesDirect examination of a magnetogram reveals a broad

spectrum of feature sizes and strengths, and suggests thatthe strongest features in the background tend also to be thelargest. Spatial autocorrelations (see ° 3.1) of individualmagnetograms are dominated by the strongest features, andthe width of the correlation peak most closely reÑects thewidths of these, rather than the mean width of all the fea-tures that are present.

The shape of the peak contains information : in Paper I itwas seen that the spatial autocorrelation peak (*T \ 0) is

remarkably triangular, which indicates, at least to theresolution of the Mount Wilson coarse arrays, that the rela-tive numbers of small features are increasingly large as theirsizes are smaller. The angular FWHM width of the peak isD 9¡ at low latitudes, suggesting that the largest featuresare D 100 Mm in longitude width. Both the shape andwidth are remarkably constant over time and nearly inde-pendent of the phase of the cycle, although there is abroadening in active region latitudes when the activity ishigh. The width increases slightly with latitude, but soslightly that, owing to the convergence of meridians towardthe poles, these largest features tend to be a little smaller inactual size at high latitudes. We shall assume the features tobe roughly ““ circular,ÏÏ as suggested from their appearanceon magnetograms, and refer to this size as their ““ diameter ÏÏin what follows.

As we have seen in ° 4, the cross-correlation amplitudesfor *T [ 0, averaged over sufficient numbers of map pairsto expose the peaks for the persistent patterns that have aconstant mean rotation rate, provide further informationabout the feature sizes, and about their relative motions anddecay. For lag times of 1 and 2 days, the peaks become lesstriangular, and by *T \ 3 days, they appear to havebecome roughly Gaussian in shape. This suggests that thesmall features have now become decorrelated. The evolu-tion of the peak shapes for lag times of three days and moresuggests features of a more uniform size which are spread-ing in time or moving randomly with respect to the meanrotation rate. As this size is intermediate between a super-granule and a plume, we shall characterize the long-livedfeatures as ““ mesoscale.ÏÏ

The same implication is present in Figure 4, where onesees from the succession of peak-height curves for increas-ing lag times (including P4 1 at *T \ 0) that the decorrel-ation is rapid at Ðrst, but then slows down. This can beunderstood if, as proposed in ° 2.3, we suppose the di†eren-tially rotating photospheric Ðeld to contain both a rapidlydecaying component and a long-lived component. If theformer is made up of smaller scale features, which duringthe lag time between correlated maps are either replaced bysimilar features in a randomly di†erent pattern, or displacedby distances large compared to their sizes, then the aver-aged correlation peak will reÑect only the latter component.

5.3. Random Walk of the Mesoscale FeaturesBoth the peak widths read from Figure 3, and the decor-

relation rates found from Figure 4 can be used to estimatemean properties and motions of the long-lived features. Toexplore this, suppose that these mesoscale features are theÐeld ““ elements ÏÏ in the SNW model, i.e., the products ofdecaying active regions, which are undergoing a randomwalk relative to the ambient di†erential rotation and pole-ward meridional Ñow. Although these features are far largerthan the Ðeld elements envisaged by SNW, we shall showthat such large sizes are necessary, given the both the modeland the observational constraints.

The random walk is characterized by a two-dimensionalGaussian probability distribution :

Õ(u, v) \ 14nD

m*T

exp[Au2] v24D

m*TB

, (1)

where (u, v) is the displacement of a feature from its initialposition during the time interval *T , and is the e†ectiveD

m


di†usion constant where L is the step size(Dm

4 0.25L2/q,and q is the step time). Let us assume that the featuresthemselves are not undergoing appreciable intrinsicchanges as they move about the Sun. This is justiÐed interms of the weak dependence on latitude of the spatialautocorrelation peaks noted in ° 5.2 ; for in the SNW model,features only reach high latitudes by migrating from lowerlatitudes, and this migration takes a long time. Hence if thefeatures themselves were spreading in time, it would showup, against the backdrop of the converging meridians, as adramatic increase with latitude in the angular width of thecorrelation peak at Ðxed lag-time. An inspection of Figure 3conÐrms that the angular longitude widths of the corre-lation peaks do not increase appreciably with latitude,except perhaps at the highest latitudes where the amplitudeis no longer clearly demarked.

We thus suppose that the increase in peak width with lagtime, seen by comparing the 3 day and 30 day curves inFigure 3, results not from intrinsic spreading, but entirelyfrom the random motions of the individual features. We canmodel this assuming an ensemble of circular Gaussianshaped features of Ðxed radius that are undergoing aa

mrandom walk on the solar surface. The assumption ofGaussian shapes makes the calculations simple, avoidingproblems noted in Mosher (1977). The normalized cross-covariance amplitude for an ensemble of widely spaced fea-tures whose displacements from their original positions isgoverned by equation (1) is then

C(L )\ 11 ] 2a

m~2D

m*T

expA [L22a

m2 ] 4D

m*TB

, (2)

where L is the lag in longitude measured from the corre-lation maximum. Assuming that equal numbers of featuresof each sign are present along a given latitude strip, themean Ðeld is zero and C(L ) is just the Pearson amplitude.We can therefore compare with Figure 3 : measuring thelongitude half-widths of the correlation ampli-W \ *L 1@etudes at low latitudes, we obtain forW3~day^ 5¡.5 ^ 0¡.5*T \ 3 days, and for *T \ 30 days,W30~day ^ 8¡.0 ^ 0¡.5at both phases of the cycle. Assuming this increase in widthto be due only to the random motions of the featuresbetween observations, we Ðnd

Dm

\ W 22[ W 124(*T2[ *T1)

\ 530 ^ 100 km2 s~1 . (3)

Solving for yields a mean feature diameteram

2am

\ 90 ^ 5Mm. This is slightly smaller than our earlier estimate, butour assumption of Gaussian shaped features gives theinferred size as the FWHM, which is slightly smaller thanthe size inferred from the Pearson amplitude, as discussed in° 4.2. Since our results are approximate, we shall continue touse 100 Mm as the typical diameter of the mesoscale fea-tures.

5.4. Decay of the Mesoscale FeaturesTurning now to Figure 4, we can use the decorrelations

to estimate the proportion of short-lived and long-lived fea-tures as a function of latitude. From this we again canestimate a di†usion constant, and can explore the decays ofthe mesoscale features. Suppose, as suggested in ° 5.2, theÐeld to consist of a short-lived ““ evaporating ÏÏ componentand a long-lived ““ wandering ÏÏ component of features that,on the timescale of the lag times *T we are considering, are

unchanged. We can then write the amplitude of the corre-lation peak at L \ 0 as

C(*T ) ^A

m1 ] b*T

] (1[ Am) exp ([*T /q) . (4)

Here is the fractional contribution to the amplitude ofAmthe long-lived Gaussian (i.e., mesoscale) features present at a

given latitude, q is the evaporation rate of the short-livedfeatures and the parameter b is to be compared with

from the model amplitude given in equation (2). In2am~2D

mwriting the lag-time dependence of the correlation ampli-tude in this way, we are ignoring the contribution to thedecrease in amplitude from the meridional Ñow out of thelatitude strip. The passage of 100 Mm features through a 6¡wide strip takes about 5 solar rotations if we assume aconstant meridional Ñow of 10 ms~1, so the contribution issmall on a timescale of a single rotation, and is e†ectivelyabsorbed into the value determined for b.

In Paper I it was estimated that the short-term decorrela-tion time q was D 1 day. A two-parameter Ðt of equation (4)to the data from Figure 4 conÐrms that this is correct, andthat q is essentially independent of latitude, which is consis-tent with small-feature lifetime estimates made elsewhere(e.g., Komm et al. 1993, 1995). The Ðt does not convergeconsistently at all latitudes, however, and since in the 30 dayand 60 curves we can ignore the short-lifetime featuresentirely, we instead use these two curves to simply solve for

and b. We then regard the corresponding 30 and 60 dayAmlagged maps as in ° 4.2 ; viz. as superpositions of maps of

random-walk-displaced Gaussian mesoscale features withrandom arrays in the proportions and respec-a

m(1 [ a

m),

tively, and determine the mesoscale feature survival param-eter using Figure 6b. According to ° 4.2, this simplya

mmeans multiplying the values we have found for by 0.75.AmWe have disregarded the polemost latitudes, for which the

correlations improve slightly, since the rotation rate here isnoisy and not reliably dominated by the DR features.

The results are seen in Figure 7. The parameter whicham,

represents the fraction of the photospheric Ñux contained inthe mesoscale features, is distinctly peaked in the active

FIG. 7.ÈProportion of long-lived mesoscale features ( Ðlled circles),amand decay parameter b as functions of latitude. The values shown are

inferred from the 30 day and 60 day correlation amplitudes seen in Fig. 4,averaged over both hemispheres.


latitudes. Its decrease with latitude, beginning at the pole-ward edge of the activity zone, is strikingly smooth.Assuming these features are only produced in activeregions, this suggests that they are losing Ñux as theymigrate to higher latitudes, even though their sizes, inferredfrom the peak width analysis, do not seem to be changing.This does not contradict our earlier assumption that thefeatures were unchanged during the lag time between corre-lated maps since the migration time scale is far longer.

The parameter b, which we might expect to have a con-stant value close to day~1, has ab

m4 2a

m~2 D

m\ 0.0036

latitude dependence similar to a. But since equation (4) doesnot provide a reasonable picture for regions where the Ðeldfeatures either are being produced, or are crowded together,this is not surprising. At very high latitudes the(Z60¡),Carrington map in Figure 1 shows that the plumes becomenearly horizontal and take on the appearance of elongateduniform unipolar features. This is where the polar crowntends to form, and this elongation must increase the e†ec-tive value for which could account for the decrease in ba

m,

near the poles. Unfortunately, the correlation amplitudes inFigure 3 become so poorly demarked by these latitudes thatwe can no longer measure the widths directly.

We can thus only expect b to have a constant value com-parable to in regions away from both active latitudesb

mand the poles. If we take the average for b over latitudes\10¡ and between 35¡ and 60¡, which is where b doesappear to be roughly constant, we Ðnd b \ 0.047^ 0.013day~1. Although the scatter is too large to make a conÐdentassertion, this is slightly larger than which might reÑectb

m,

the additional e†ects of both meridional Ñow and decay ofthe features.

It seems likely that the decay of the mesoscale featuresalso involves a di†usive process, since high-resolutionobservations reveal all magnetic regions on the solar surfaceto be aggregates of tiny Ñux elements, which are movingabout locally as if on little random walks propelled by thegranular and supergranual di†usion. If this is so, thedecrease in the peak Ðelds of these features, still assumingGaussian shapes, would be given by

Fm(T )\ F

m(0)/(1 ] 2b

sT ) , (5)

where and are the maximum strengths theseFm(0) F

m(T )

features have when created at T \ 0 in the active regionsand later when they have reached some other latitude, andwhere with the e†ective di†usion constantb

s\ 2a

m~2 D

s, D

sfor the small-scale random walk of the individual Ñux ele-ments.

The di†usive decay of a Gaussian also involves broaden-ing. In our analysis of Figure 3, however, we had to attrib-ute the broadening of the correlation peak with lag time tothe random walk displacements of the features betweenobservations rather than to a broadening of the featuresthemselves. If the mesoscale features are not broadeningwith time, then as they decay, their di†usive widening mustbe countered by Ñux loss through reconnection and evapo-ration along their borders.

To determine the rate of Ñux loss in the mesoscale fea-tures, we obtain their Ñux as a function of latitude by multi-plying their relative contribution by the total Ñux. Usinga

mthe distribution by latitude of total Ñux given in Howard &LaBonte (1981), we Ðnd, as seen in Figure 8, an eight-folddecrease in Ðeld strength between latitudes of 35¡ and 65¡.But if km2 s~1, equation (5) would thenD

s\ D

m\ 530

FIG. 8.ÈHigh-latitude values for the mean Ðeld strengths of theFmmesoscale features. The solid line is the best Ðt assuming Gaussian features

originating at latitudes ¹ 35¡, and moving about with a combination of 10ms~1 poleward meridional Ñow and a random walk with e†ective di†usionconstant km2 s~1. This Ðt gives a di†usion constant for theD

m\ 530

decay of the features to be km2 s~1.Ds\ 250

imply a transit time less than 100 days, which is obviouslytoo short. We can estimate the time T it takes a typicalfeature to move a latitudinal distance *h : if we assume acombination of random walk with our previously deter-mined and meridional Ñow, which to simplify the argu-D

mment, we overestimate as constant at a rate ms~1,vmf

\ 10T is found by solving Rsun*h [ v

mfT \ 0.832(4D

mT )1@2.

For our di†usion constant km2 s~1, *h measuredDm

\ 530in degrees latitude and T in measured in days, this gives

T \ 85.4(1] 0.162*h[ J1 ] 0.325*h) . (6)

For a transit of *h\ 30¡, T ^ 230 days. Since we haveoverestimated this is a lower bound on the mean timev

mf,

for this migration.The decrease with latitude of the proportional contribu-

tion of the mesoscale features is too gradual, therefore, fortheir disintegration to be consistent with Di†usionD

s\ D

m.

constants that have been found for small-scale featuremotions motion range from 250 km2 s~1 from Big Bear Ðnescans (Schrijver & Martin 1990) to 285 km2 s~1 from mea-surements on SOHO/MDI magnetograms (Hagenaar et al.1999). We can estimate our from Figure 8, using equa-D

stion (6) to express as Since Figure 4 was aFm(T ) F

m(*x).

full-cycle average, the active regions range up to D 35¡, andwhen we Ðt determined from equations (5) andF

m(x[ 35¡),

(6) to the data for latitudes higher than 35¡, we obtain thesolid curve plotted in Figure 8. As the Ðgure shows, this Ðt isquite good, and we obtain day~1. This gives ab

s\ 0.017

di†usion constant km2 s~1, in agreement withDs^ 250

high-resolution measurements.

6. SUMMARY AND CONCLUSIONS

We have examined properties and motions of features inthe photospheric background magnetic Ðeld that can bededuced from one-dimensional correlations. With regard tothe rotation of the Sun as determined by the passage ofthese features across the solar disk, we have veriÐed thatpatterns of both di†erential and rigid rotation are present.


We have found, at the scale of the Mount Wilson magneto-grams, evidence for two types of magnetic features thatexhibit di†erential rotation like that of the photosphericplasmaÈshort lived features that decorrelate on a timescaleof roughly a day, which are presumably the type found byKomm et al. (1995), and thus smaller than a supergranule,and long-lived features that persist for more than a solarrotation and have mean diameters roughly 4 times thesupergranule diameter. The latter are thus probably moredeeply rooted, which is consistent with their slightly fasterrotation rate, noted in ° 2.3.

Our results are in part consistent with the models ofLeighton (1964) and SNW, in that we can attribute the rigidrotation patterns to the random walk and meridional Ñowof the di†erentially rotating individual features. We haveshown, however, that the correlation-determined proÐle ofrotation as a function of latitude depends more on themanner in which the calculation is done than on the lengthor time scales that are employed.

The decay with lag time (i.e., time separation betweencorrelated maps) of the peak amplitude of the cross-correlation, which is at Ðrst rapid, as was seen in Paper I,but slows dramatically after a few days, implies two dispa-rate decorrelation rates. In the correlation amplitudes atlong lag times, the contribution of the short-lived Ðeld ele-ments is e†ectively Ðltered out, and this has enabled ourstudy of the long-lived features. Our main inferences are thefollowing :

1. The mean diameters of the long-lived, ““ mesoscale ÏÏfeatures do not vary signiÐcantly with latitude, and are^100 Mm. These features are far larger than was assumedin WS, which accounts for the long lag-time persistence ofthe DR peak in the correlation amplitude.

2. If we assume, as in the SNW model, that the long-livedfeatures are produced in active region latitudes, and subse-quently roam about in a random walk with a polewardmeridional drift, we obtain a di†usion constant D

m\ 530

^ 100 km2 s~1. This agrees favorably with the valuerequired in the SNW model to simulate the polar Ðeldreversals of cycle 21.

3. The long-lived features do not spread in time, but thenet Ñux contained within them decreases with time. If wesuppose this Ñux decrease involves a di†usive process, thedi†usion constant is less than We obtain the valueD

m.

km2 s~1, which is consistent with high-resolutionDs^ 250

studies of the motions of individual subarcsec Ñux elements.This suggests that the long-lived features are aggregates ofthe subarcsec elements, and that their decay is taking placethrough the small-scale random walks, recombinations, andevaporation of these elements.

The logic behind these inferences deserves a briefsummary. In studying the long-lived features, we utilizedboth dependencies of the cross-correlation amplitude : as afunction of lag-time *T at Ðxed latitude, and as a functionof latitude at Ðxed lag-time. Adopting the SNW model, thelatitude dependence in the latter was translated to a roughmeasure of elapsed time T since leaving active latitudes. Wethus could look at both feature and pattern decay, for whilethe dependence on *T gives information on the combinede†ects of both, the dependence on T itself, extrapolated to*T \ 0, is connected only to the attributes of the individualfeatures. While the measure of T is admittedly imprecise, itis sufficient to show that these features cannot be loosing

Ñux as rapidly as they would if their decay were governedby the SNW di†usion constant.

If the mesoscale features are, as suggested by theirdecays, aggregates of tiny random-walking Ðeld elements,then they should be broadening in time by a factorD(1] 0.017T )1@2, so that by the time an aggregate reachesD60¡ latitude its width should double. Since such spreadingwith latitude is not seen in the correlation peak widths, Ñuxmust be lost by reconnection and/or evaporation. A rapidevaporation of the bulk of the recognizable Ñux, whichimplies that new Ñux with a similar scale of feature sizes andstrengths, must appear to replenish it, has long been noted(e.g., Howard & LaBonte 1981). Though our two-component picture is obviously oversimpliÐed, Figure 7also suggests this, giving the proportion of short-lived fea-tures to run to more than 90% near the poles.(1 [ a

m)

Note, however, that the values of a determined in the activelatitudes cannot be trusted since the determination assumesthat the density of mesoscale features is low, and that merid-ional Ñow, random walk and decay are the only processes.

In order that the features widths and amplitudes remainconstant in time at a given latitude, the features must eitherbe replenished locally or must move across latitudes as theyevolve. If the plumes, which exhibit rigid rotation are indeedcomposed of bundles of Ñux that have separated from theactive regions and are undergoing random walk and pole-ward meridional Ñow, then these bundles could only be thelong-lived features. The short-lived features, on the otherhand, must be created locally and could only contribute tothe plumes if their density and orientation were a†ected bylocal processes associated with the plumes (Snodgrass &Wilson 1993) or with the long-lived features.

Thus the results of our correlation analysis can be inter-preted as a partial conÐrmation of the SNW model. WeÐnd, however, that the features which create the large-scale,rigidly rotating plumes by undergoing the prescribed com-bination of motions are themselves far larger than antici-pated in the model. Their surface areas span D 15supergranules, and while their individual decays may be thework of granular and supergranular convection, this cannotbe what propels their random walk. Collective behavior ofthe supergranulation at this scale has not been observed,and the randomness of the convective motions within apatch of 15 such cells would tend to dampen the netimpulse. We conclude that the mesoscale features must berooted below this tier of convection, and their apparentrandom walk at the sunÏs surface must result from somelarger-scale process.

We emphasize that the determination of the size of thesefeatures, which was done through an analysis of the widthsof the correlation peaks, is conÐrmed through the longtime-lag persistence of the peak corresponding to di†eren-tial rotation. For if we agree that these features are onrandom walks and suppose, for a lower bound, they are notchanging intrinsically, the decay rate of the correlationpeak, given in equation (2), depends on the ratio D/a2. Withan e†ective SNW di†usion constant DD 600 km2 s~1, adiameter of 2a D 100 Mm is required for the persistenceseen in Figure 4, which is 1 order of magnitude longer thanthat found in the WS simulation.

The existence, apparent motions and decay of the meso-scale features thus raise interesting questions. Two-dimensional correlations track these same mesoscalefeatures and Ðnd a cycle-dependent meridional Ñow which


is directed away from the active-region latitudes (Snodgrass& Dailey 1996). A question being explored in a separatepaper is whether this Ñow pattern itself is consistent withFickÏs law for the SNW prescription of random walk plusmeridional Ñow. The results presented in this paper arepreliminary ; a more thorough exploration of the evolutionof the Ðeld patterns over the cycle, using the techniquesdeveloped in this paper, is being planned.

We gratefully acknowledge R. K. Ulrich, John Boyden,and the observing sta† at Mount Wilson Observatory forsupplying the magnetograph data. We also acknowledgehelpful discussions with Neil Sheeley, Jr., Peter Wilson, andwith members of the sta† at Big Bear Observatory. Thiswork was supported by grants AST94-16999 and ATM98-14145.

APPENDIX A

CHOICE OF CORRELATION FUNCTION

Functional deÐnitions associated with correlation theory vary with the source. We follow Jenkins & Watts (1968), andbegin by deÐning the cross-covariance by

Cfg

(L ) 4PA

Bf (/)g(/] L )d/ , (A1)

in which f (/) and g(/) are Ðelds and L is called the lag. In our application to determining the rotation of the Sun, f and g arethe magnetic Ðelds as a function of longitude at a speciÐc latitude at two di†erent times. The longitude / may either beCarrington longitude, or longitude relative to a Ðxed meridian on the solar disk. The covariance amplitude C(L ) has itsmaximum at lag i.e., at the displacement of the pattern through the solar rotation. This follows upon integration byL \ L

M,

parts, when (1) the Ðeld does not change intrinsically with time, and (2) is either periodic on [A, B], or else the domain [A, B]\ [[O,O]. The function C(L ) is symmetric about this maximum, hence a curve Ðtting routine can be used to determine L

M.

Random intrinsic changes in the Ðeld reduce the peak amplitude, but have a random e†ect on the LM

.Real data consist of Ðnite numbers of measured values. Thus the integral in equation (A1) must be replaced by a sum, the

lag L is stepped in intervals */ equal to the bin size, and the sum can only include terms where both f (/) and g(/] L ) aredetermined. Thus for a longitude range [a, b] over which the Ðelds are measured which is the same size for both f and g, thesum can only include terms for which / lies within the interval [A, B], where

[A, B]U [a, b [ L ] for L [ 0, and

[A, B]U [a ] L , b] for L \ 0 . (A2)

This may be done either by summing over a moving subwindow of Ðxed width W \ B[ A determined from the maximumlag to be employed, or by summing over the maximum window for each lag, which shrinks as indicated in equation (A2) as themagnitude of the lag increases.

For either case, however, determined from equation (A1) no longer necessarily gives the true pattern displacement, andLMthe covariance amplitude C(L ) is no longer symmetric about Integration by parts when W is held Ðxed shows that theL

M.

maximum equals the pattern displacement only when f (A)g(A] L ) \ f (B)g(B] L ) for L in the neighborhood of henceLM

;for periodic Ðelds with period W this approach would seem appropriate. Although the Ðelds in our application are periodicfor W \ 2n, the measured Ðelds are not periodic, since they change in time and the whole 360¡ is not observable at once.

Shrinking the longitude interval with increasing o L o uses the data as fully as possible, but C(L ) is now biased toward smalllags. The bias is greatly reduced, however, when instead of the cross-covariance of equation (A2), the cross-correlation, whichis the cross-covariance of the normalized Ðelds, is used, and this is the approach we adopt. We subtract, furthermore, the meanvalues, which gives zero correlation if either f (/) or is constant over the domain window. The resultingg(/] L ) 4 g

L(/)

correlation function is known as the Pearson correlation. To wit :

Pfg

(L ) 4 CF4 G4

(L ) . (A3)

Here refers to the normalized function where and similarly, refers toFŒ FŒ (/)\ F(/)/(CFF

(0))1@2, F(/) 4 f (/) [ S f T*A,B+, GŒwhere with S indicating the average over theGŒ (/] L )\ G

L(/)/(C

GL GL(0))1@2, G(/] L )4 G

L(/) 4 g

L(/) [ Sg

LT*A,B+, T*A,B+domain window. Both the averagings and the normalizations are done over the domain window appropriate to each speciÐc

lag L according to equation (A2).The properties of both C(L ) and P(L ) become obvious when one views them algebraically. The binned Ðeld values may be

regarded as components of a vector on the Ðnite-dimensional vector space of dimension N \ (b [ a [ o L o )/w, where w is thebinwidth (cf. eq. [A2]). The cross-covariance is the inner product of two vectors f and on this space, and itsC

fg(L ) g

Lmagnitude depends on both their alignment and their magnitudes. One cannot assert, therefore, that the maximum of Cfg

(L )will occur when the alignment is optimal, i.e., when the lag equals the pattern displacement. On the other hand, is theP

fg(L )

inner product to two unit vectors, and thus has its maximum value P(L ) \ 1 only when they are perfectly aligned. Despite thefact that the dimension of the space changes with changing L , the lag of the maximum in P(L ) must therefore, disregardingintrinsic changes in the Ðeld, give the pattern displacement to within the accuracy of the binning interval.

The correlation peak is not symmetric about its maximum for either C(L )or P(L ). The asymmetry is mostly random, butthe extraction of the means in the Pearson correlation leads to a slight systematic bias. To see this, consider a function f (/)which is zero everywhere except for a narrow range of values a \/\ a] v. Let Then C(L ) is zero exceptg(/) \ f (/ [ /0).


for a symmetric peak of width v about and curve Ðtting will give On the other hand, the subtraction of theL \/0, LM

\/0.means causes P(L ) to be negative, suggesting anticorrelation, everywhere except at the peak, which is again at TheL \/0.peak is not symmetric because the extraction of the mean values for f and g lowers the values of F(/) and G(/) by increasingamounts as the domain window over which the mean is computed shrinks. Thus, for L [ 0, P(L [ v) [ P(L ] v), and curveÐtting will give underestimating the true displacement. Numerical experiments show that this error is small (lessL

M\ /0,

than 0.1% in practice), and that its disadvantage is o†set by the fact that the removal of the mean causes the correlation peakto more closely match the sizes of the features than is the case when the means are not extracted.

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