Evolution and motions of magnetic fragments during the active region formation and decay: A statistical study
AAstronomy & Astrophysics manuscript no. 40127 © ESO 2021February 5, 2021
Evolution and motions of magnetic fragments during the activeregion formation and decay: A statistical study
Michal Švanda , , Michal Sobotka , Lucia Mravcová , , and Tatiana Výbošt’oková Charles University, Astronomical Institute, V Holešoviˇckách 2, CZ-18000, Prague 8, Czech Republic Astronomical Institute of the Czech Academy of Sciences, Friˇcova 298, CZ-25165 Ondˇrejov, Czech Republic Charles University, Department of Surface and Plasma Science, V Holešoviˇckách 2, CZ-18000, Prague 8, Czech RepublicReceived date / Accepted date
ABSTRACT
Context.
The evolution of solar active regions is still not fully understood. The growth and decay of active regions have mostly beenstudied in case-by-case studies.
Aims.
Instead of studying the evolution of active regions case by case, we performed a large-scale statistical study to find indicationsfor the statistically most frequent scenario.
Methods.
We studied a large sample of active regions recorded by the Helioseismic and Magnetic Imager instrument. The samplewas split into two groups: forming (367 members) and decaying (679 members) active regions. We tracked individual dark features(i.e. those that are assumed to be intensity counterparts of magnetised fragments from small objects to proper sunspots) and followedtheir evolution. We investigated the statistically most often locations of fragment merging and splitting as well as their properties.
Results.
Our results confirm that statistically, sunspots form by merging events of smaller fragments. The coalescence process isdriven by turbulent di ff usion in a process similar to random-walk, where supergranular flows seem to play an important role. Thenumber of appearing fragments does not seem to significantly correlate with the number of sunspots formed. The formation seems tobe consistent with the magnetic field accumulation. Statistically, the merging occurs most often between a large and a much smallerobject. The decay of the active region seems to take place preferably by a process similar to the erosion. Key words.
Sun: Sunspots, Sun: Activity, Sun: Magnetic fields
1. Introduction
Strong local magnetic fields forming active regions are the mainmanifestations of solar magnetic activity. The formation, evolu-tion, and structure of these active regions have been studied formore than four centuries; however, they are still not completelyunderstood. Reliable information about the structure of active re-gions beneath the surface does not exist (Moradi et al. 2010). Al-though there were attempts to use local helioseismology to inferthe depth structure of sunspots (e.g. Zhao et al. 2001), they werelargely questioned recently (e.g. Gizon et al. 2009). Thus all ofour useful knowledge about the 3D structure of, for example,sunspots and their surroundings comes from theoretical models(e.g. Rempel et al. 2009) or topological studies (e.g. Chintzoglou& Zhang 2013).The current paradigm is that solar magnetic fields are gen-erated by dynamo action deep in the convection zone. Whenthe magnetic field is su ffi ciently strong, it becomes buoyant andemerges in the form of toroidal flux ropes oriented in the east-west direction, forming bipolar active regions on the surface.The emergence of the flux alternates the mode of convection,which is mostly represented by granular cells at the surface. Thiswas studied by Cheung et al. (2007), for example, in a numeri-cal simulation. They concluded that due to the high Mach num-bers found in convective downflows, it is virtually impossible forbuoyant flux tubes to rise unimpeded by the convection. Send o ff print requests to : M. Švanda, e-mail: [email protected] As a result of the very vigorous convection near the solarsurface, the morphology of emerging flux regions will be highlystructured by the near-surface convection. A general overviewof the evolution of active regions was delivered by van Driel-Gesztelyi & Green (2015), where the authors, in general, de-scribe the processes considered during the active regions’ life-time starting from the pre-emergence phases and ending by fad-ing the enhanced network to join the background magnetic field.Despite the very thorough description, the authors conclude thatnot all the phases of the active regions’ evolution are understoodsu ffi ciently.Unfortunately, lately, specific phases of the evolution, themorphology of active regions, as well as large sunspots namelyhave not drawn the attention required. For instance, accordingto the classical paper by Zwaan (1985), the sunspots form bycoalescence of a few larger fragments, which are themselvesmade of typically two-three fragments. According to Garcia deLa Rosa (1987), these fragments retain their identity duringthe whole evolution of sunspots, even though they change theirsize, shape, and position. When the sunspot starts to decay, itsbreak-up starts at the boundaries of those fragments, and lightbridges usually form there. Some of these fragments survive thesunspot’s break-up and turn in the long-lived spots of simplermagnetic configuration of late Zürich type. Such a picture de-serves confirmation in a statistical sense using state-of-the-artdata.On the other hand, decaying sunspots with penumbra areusually surrounded by a moat – a region around the sunspot,where the flows prevail radially out from the sunspot (Sheeley Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . S R ] F e b & A proofs: manuscript no. 40127 ff erent scenario than thedecay by large fragments. The formations or decays of sunspotswere typically studied on a case-by-case basis (e.g. Chicralaet al. 2017), mostly using campaign high-resolution data (e.g.Kontogiannis et al. 2020). These processes were also targets ofnumerical simulations.State-of-the-art numerical simulations show the sunspot evo-lution from the modellers point of view. For example, Cheunget al. (2010) showed that when the bipolar active region emerges,first, small-scale magnetic elements appear at the surface. Theygradually coalesce into larger magnetic concentrations, whicheventually results in the formation of a pair of opposite polar-ity spots. The whole active region evolution was simulated byRempel & Cheung (2014); the description includes the decayof the active region, which was driven by the flows from thesub-surface. An almost field-free plasma intruded the magneticfield and when these intrusions reached the photosphere, the spotfragmented. Dispersal of the flux from the simulated sunspot wasconsistent with the decay by turbulent di ff usion.Similarly, Chen et al. (2017) studied the emergence of a mag-netic flux bundle from the convection zone to the corona. Thenumerical simulation among others resulted in the synthetic con-tinuum intensity images that were equivalent to high-resolutionobservations. The simulation shows that the magnetic flux bun-dle rose as a coherent structure throughout the upper convec-tion zone, except for the uppermost several megametres, wherethe bundle fragmented. These fragments consisted of magneticflux tubes emerging individually at the surface. The syntheticwhite-light images showed that first, small granular-sized mag-netic elements appeared at the surface and they gradually coa-lesced to pores. As the emergence continued, the pores mergedto sunspots.It would seem that coalescence of small-scale magnetic fea-tures with the formation of larger structures is dominant duringthe magnetic flux emergence (this process is referred to as in-verse turbulent cascading, where the energy transfers from smallscales to larger scales). The full-scale inverse cascade was ob-servationally reported for the first time by Hewett et al. (2008)when studying a sequence of magnetograms of a particularly fastemerging active region NOAA 10488.The emergence takes place in a very turbulent medium inthe upper solar convection zone, so the opposite process occursat the same time. This process is referred to as the direct cas-cade when the magnetic structures naturally fragment to smallerones. The interplay of both cascades during the emergence ofseveral active regions was confirmed by Kutsenko et al. (2019).They conclude that most of the time the energy grew at all scales.The study could not state that the inverse cascade was a domi-nant process during the formation of an active region. Althoughcoalescence of small magnetic elements into larger pores andsunspots was observed, in terms of energy contribution to theactive region the emergence of large-scale structures was moreimportant.The availability of the synoptic observations with publicavailable archives increased the number of captured sunspot for-mations and decays. In the modern-most synoptic observationsby the Helioseismic and Magnetic Imager (HMI; Scherrer et al.2012) on board of the Solar Dynamics Observatory (Pesnell et al. 2012), the spatial sampling of 0.5" is su ffi cient enough to resolvesmall magnetic features from which sunspots are supposed to beformed by coalescence according to the scenario of Garcia de LaRosa. The long-term coverage of over 10 years with a duty cy-cle close to 100% allows for one to select suitable active regionsrecorded during these critical phases of their evolution. Insteadof studying their evolution case by case, we performed a large-scale statistical study to find indications for a general scenario.This general or most frequent scenario does not preclude otherdi ff erent possibilities. Nevertheless, these di ff erent possibilitiesare statistically less likely.
2. Data
From the HMI archive of pseudocontinuum intensitygrams (dataseries hmi.Ic_45s ), we selected sunspots having the followingproperties.For the forming active regions we required that (1) an activeregion emerged not farther than about 60 degrees from the disccentre and (2) it survived within this distance for at least 3 days.For further analysis, we considered all of the days when the ac-tive region was within the central meridian distance (CMD) of60 degrees. Additionally, we also considered 2 days before theemergence regardless of the CMD of the prospective active re-gion location. In total, a sample of 367 members was identified.For decaying active regions we required that (1) an activeregion decayed not farther than about 60 degrees from the disccentre and (2) before the final decay, it was observable for at least3 days within the distance of 60 degrees from the central merid-ian. For further analysis, we considered all of the days when theactive region was within the CMD of 60 degrees. Furthermore,we also considered one additional day after the final decay re-gardless of the CMD of the past active region’s location. In total,679 members were identified in the data archives.The selection of the appropriate active regions was donemanually closely cooperating with NASA’s SolarMonitor . Thetime range covered was between May 2010 and August 2018.The dismemberment into a forming or decaying group was donesubjectively following the simplest possible criteria: When theactive region emerged within a CMD of 60 degrees, it was con-sidered for the group of forming active regions. Similarly, whenthe active region finally decayed (i.e. no sunspots were observedon the following days) within a CMD of 60 degrees, it was con-sidered for the group of decaying active regions. The days beforethe emergence or after the final decay, respectively, were con-sidered as the safety margins to make sure we did not miss theappearance of possible small ephemeral objects on those days.For each group, we tracked the datacube using the Carring-ton rotation and selected the field of view having 768 ×
768 pix-els centred on the AR coordinates given by SolarMonitor. Thedatacube consisted of two spatial coordinates (one correspond-ing to the zonal east-west direction and the other to the merid-ional south-north direction) and one time coordinate. The frameswere Postel projected with a pixel size at the centre of the fieldof view of 0.0301 heliographic degrees, which corresponds to0.366 Mm. The frame cadence was 12 minutes, which was atrade-o ff between the data volume and the time resolution. Thetracking and mapping were done using the JSOC tools.The missing frames were linearly interpolated so that the dat-acube had a regular sampling in time. For each considered dat- Joint Science Operation Centre, jsoc.stanford.edu
Article number, page 2 of 14vanda et al.: Evolution of magnetic fragments in active regions acube, we also created a visual movie for the inspection of theevolution by eye.
3. Methods
Henceforth, the terms ‘object’ and ‘fragment’ indicate the samequantity, that is a dark object detected by the pipeline describedin this section. The pipeline is based on the object tracking algo-rithm developed by Sobotka et al. (1997) for tracking small-scalefeatures in sunspots.In the first step, continuum-intensity frames that include theedge of the solar disc, appearing at the beginning and / or end ofthe time series, are discarded. Then, the centre-to-limb inten-sity gradient is removed to obtain flattened frames with pho-tospheric intensity I ph normalised to unity. The normalisationvalue is obtained by the flattened field of view averaging witha clearly dominant contribution of the quiet photosphere. Simpleimage segmentation is applied to these frames, defining objectsas groups of pixels with intensities of I ph < .
9. Side-by-sideneighbouring pixels form the same object. Single-pixel objectsare considered as noise and discarded. The objects retain theiroriginal intensity, while the intensity in the rest of the field ofview is set to zero.In the second step, evolutionary histories of objects aretracked in a series of segmented frames. An object continues itshistory if at least one of its pixels coincides in position with anobject’s predecessor in the preceding frame. There are four pos-sibilities for the object-related event: the object appears, it mayfade out, it may merge with another object, or it may split intwo. The rules applied for merging and splitting are describedin Section 3.3. The code records the area-averaged intensity, thecoordinates of the centre of gravity, and the number of pixels(area) of each object for each time instant (frame) of its history.In the tracking results, objects with time-averaged areas smallerthan nine pixels, that is to say 1.2 Mm which is comparable toa large granule, are considered noise and discarded.In the third step, the tracking results are used to map eventsof merging and splitting. For the instant of birth or death ofa tracked object, the code looks for a preceding or successiveneighbour object, from or to which the tracked object can splitor merge (see Section 3.3). When such an object is found, infor-mation about the two objects, time, and the type of event (splitor merge) is recorded. The object detection pipeline detects all objects that have an in-tensity below a given threshold. That also includes sunspots. Itis not straightforward to distinguish sunspots from other objects(fragments) using just a simple criterion. On the other hand, todeal with the science goals of our study, the need for a referenceseries containing only detected sunspots is perspicuous.To detect only sunspots, we chose to follow the methodologyalready published in the literature. We used an object identifica-tion algorithm based on the mathematical morphology, follow-ing the procedure described by Watson et al. (2009), and codedusing Python’s Scikit-Image library.The continuum datacube was processed frame-by-frame. Foreach frame, we applied the median filter (with a boxcar of 20px) to remove the small-scale noise such as the cosmic-ray hitsor very small features. Then we applied a morphological top-hat transform (Dougherty & Lotufo 2003) to image I . Top-hat trans-forms are used for various image processing tasks, such as fea-ture extraction, background equalisation, image enhancement,and others. We used the white top-hat transform, which is de-fined as the di ff erence between the input image and its openingby some structuring element. The structuring element must belarger than the features in order to be detected. As a structuringelement A , we used the circle having a diameter of 196 px, whichcorresponds to about 72 Mm.The white top-hat transform is then defined by T w ( I ) = I − I ◦ A , (1)where ◦ indicates the morphological opening operation, whichcorresponds to an erosion followed by a dilation. We note thatthe white top-hat transform is designed to extract brighter fea-tures, so in order to extract dark sunspots, one has to invert image I first. A nice example of the individual steps of the procedure isgiven in Fig. 1 in Watson et al. (2009).All pixels of T w ( I ) larger than 6000 DN / s (an experimen-tal threshold) are considered as sunspots . Automatic labellingof continuous regions was performed to identify individualsunspots. Due to the median filter, only sunspots larger than afew megametres were detected. Small pores avoid the detectionbut they were retained as fragments in the fragment detectionpipeline.The thresholds of both detection pipelines were set such thatany feature at least 10% darker than the quiet photosphere wasidentified as a separate object. That is, umbrae and penumbraefall below the thresholds. In the case of evolved sunspot, it isidentified as one compact object including the penumbra sur-rounding the umbra as long as they are not completely separatedby a bright structure corresponding to the quiet-Sun photospherebrightness. The very small and short-lived features are consid-ered to be noise and removed by the application of the size cri-teria. This application certainly also removes some real, short-lived small objects. We believe that this omission does not a ff ectour analysis significantly and that the benefits of discarding thenoise outweigh the loss of a certain object population.In about 10% of the active regions in our sample, no propersunspots were detected by the sunspot detection pipeline. Theseactive regions are only populated by small fragments. Naturally,in those active regions, the number of detected objects is alsomuch lower. In total, the objects contained in the active regionswithout sunspots represent 1.1% of the total. Therefore we con-sider any possible influences of these spotless active regions tothe results and conclusions negligible. The object tracking algorithm allows one to build a time-spacetrajectory and identify possible merging or splitting events. Themerging event is considered if a single object in a given frameoverlaps the pixels of two objects (at least one pixel each) inthe preceding frame. It means that two objects converged to acommon location. The follow-up object is considered to be amerger and keeps the label of the older merging object. A short note regarding the units: In Section 3.1 the algorithm utilisedflattened intensity images, which were normalised to the quiet-Sun in-tensity. In the case of Section 3.2, we kept the HMI I c units ‘data num-ber per second. Sunspots areas segmented by thresholding the di ff er-ence between the image and its background obtained by the applicationof the opening operator exceeding the threshold. Hence the value of thethreshold keeps the data units. Article number, page 3 of 14 & A proofs: manuscript no. 40127
140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ]
140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ] T=56.40 hours140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ]
140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ] T=57.80 hours140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ]
140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ] T=59.80 hours140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ]
140 150 160 170 180 190 200x [Mm]130135140145150155160 y [ Mm ] T=63.20 hours
Fig. 1.
Example snapshots from fragment detection and tracking. In the left column, one can see the cut-out from the full field of view. In the rightcolumn, the detected fragments passing the selection criteria are shown; their identity is indicated by colours. None of the detected objects overthe field of view were classified as a sunspot by the sunspot detection pipeline. There are many dark features in the pseudo-continuum images thatare not marked as objects. These unmarked features did not meet the criteria for the lifetime or size.
An example of object merging is given in Fig. 1. A demon-strable merging occurs around coordinates x =
180 Mm and y =
150 Mm. At T = .
40 hours, the first of the followed frag-ments (red) is born; at T = .
60 hours, the second one is born(purple). At T = .
80 hours, the fragments are stable, growingand travelling ahead of each other. The time T = .
20 hoursis the last frame when the fragments were separated; in the nextframe, they merge into one object, which kept the identity ofthe ‘red’ fragment. We note that the blue-grey object on co- ordinates x ∼
180 Mm and y ∼
140 Mm, which emergedat T = .
80 hours (not displayed) grew in intensity between T = .
40 hours and T = .
60 hours, then it started to fadeout and disappeared at T = .
40 hours (not displayed) withoutmerging.The splitting event is considered if two objects in a givenframe overlap at least one pixel each with a single object in thepreceding frame. This means that a new object is born just nextto a previously existing one. The label of the mother object is
Article number, page 4 of 14vanda et al.: Evolution of magnetic fragments in active regions
140 160 180 200 220 240x [Mm]110120130140150160170 y [ M m ] M e a n c o n t i nuu m i n t e n s i t y [ D N / s ] Fig. 2.
Fragments in AR NOAA 11076. The background image consistsof an intensitygram averaged over the whole datacube time span. Thewhole considered field of view is plotted. The tracks of individual frag-ments (green lines) and the positions of the merging (red) and splitting(blue) events are overplotted. kept by the darker of the daughter objects and the brighter one isgiven a new label.The merging and splitting events are recorded to a table,which contains information about the two objects considered,the time when the event occurred, and the type of the event. Thisallows one to create a spatio-temporal map of the merging andsplitting centres. One may also draw a historic trajectory of eachobject from its birth to its decay or merging. An example is givenin Fig. 2. Here one can see that most of the features are detectedwithin the dark, that is, magnetised regions.In principle, the merging and splitting events may occur any-where in the active region. However, if the scenario by Garcia deLa Rosa is valid, one would anticipate that merging and splittingevents occur around preferred locations, likely near the locationsof sunspots. To investigate this issue, we applied a cluster analy-sis to the locations of the merging and splitting events.We utilised the density-based spatial clustering of applica-tions with noise (DBSCAN, Ester et al. 1996) implementedwithin Python’s Scikit-Learn library. DBSCAN is a density-based non-parametric clustering algorithm. Given a set of pointsin some space, it groups together points that are closely packedtogether (points with many nearby neighbours). Points lyingalone in low-density regions (with nearest neighbours far away)are marked as noise. The advantage of the DBSCAN algorithmis that the expected number of clusters does not have to be knowna priori. The algorithm can also find arbitrarily shaped clusters.For each datacube, we identified the clustering of the merg-ing and splitting events. We used the minimum distance of thecluster members ε =
10 px and the minimum number of clustermembers of 2. As a distance measure, we used the 2D Euclidiannorm. We did not consider a factor of time in the distance met-ric; therefore, the splitting and merging events belonging to thesame identified cluster did not have to occur at the same time.An example of the identified clustering is given in Fig. 3.This is a typical example where one can see that most of thesplitting and merging events are located within the regions withsunspots.
140 160 180 200 220 240x [Mm]110120130140150160170 y [ M m ] Estimated number of clusters: 16
Fig. 3.
Clustering of merging and splitting events over the portion ofthe field of view of NOAA 11076. Di ff erent colours indicate detectedclusters. Black points with white strokes represent points which do notbelong to any of detected clusters.
4. Results
For each active region, we obtained several data products asfunctions of time. Firstly, the number of detected individualsunspots, in case there are multiple umbrae within one penum-bra, they count as one sunspot. Secondly, we measured the totalarea taken by the detected objects. Thirdly, we put together anoverview of the fragments in each frame including (a) newly ap-pearing fragments, (b) fragments resulting from the splitting ofthe mother fragment, (c) fragments disappearing because theymerged with other fragments, and (c) fragments that faded stay-ing alone. These quantities were evaluated not only by the framebut also in their cumulative forms, that is the sum of the quan-tity from the beginning of the sequence to the given timestampmatching the stamp of the given frame.An example for one active region NOAA 11076, where wecaptured not only its formation but also a significant part of itsdecay, is shown in Fig. 4. The region reached its maximum areaat a time of T ∼
120 hours (red line, top middle panel). Be-fore this time, merging events (negative) were dominant overthe active region (olive line, top right panel); after this moment,the splitting (positive) events prevailed. The total number of de-tected objects in the field of view reached a local minimum at T ∼
120 hours, whereas the maximum number of objects wascounted about 30 hours later. The slope of the olive curve indi-cates that between these two instances, the fragmentation of theobjects was particularly fast.Interestingly, the total number of merging events did notequal the total number of splitting events since the olive linein Fig. 4 remained negative. The remaining objects faded with-out further fragmentation. On the other hand, at each moment,the total number of appearing objects (new objects and objectsresulting from splitting events) and disappearing objects (fadedobjects and mergers) were in a very close balance (bottom mid-dle panel of Fig. 4). This would indicate a relatively short ex-pected lifetime of the objects. The object appears and soon aftermerges or fades.
Article number, page 5 of 14 & A proofs: manuscript no. 40127 a) b) Total object area [Mm ] 0 50 100 150 200T [hours]302520151050 c) Splitting (+) / merging (-) cumulative0 50 100 150 200T [hours]050100150200250300 d) e) Cummulative increment / decrementincrementdecrement 0 50 100 150 200T [hours]05101520 f) Fig. 4.
Time evolution of various quantities describing the sunspots and fragments in NOAA 11076. One can see the evolution of the number ofdetected sunspots (a) and their area (b). The cumulative curve of the splitting or merging events is also plotted (c), showing which of the quantitiesprevailed until the given moment. In panel d), one can see the cumulative curves of the number of objects that newly appeared and that fadedstaying alone. In panel e), one can see the cumulative balance between the increase and decrease in the number of objects. Finally, in panel f), thetotal number of the detected objects at the given time is plotted. ]0.000.050.100.150.200.250.30 F r e q u e n c y [ n o r m a li s e d ] F r e q u e n c y [ n o r m a li s e d ] Fig. 5.
Left: Histogram of the object sizes (in terms of the areas) computed for the binning of 1 Mm . The inset shows the magnified histogram forsmall sizes with the bin size of 0.267 Mm , which is twice the natural sampling. Right: Histogram of the object lifetimes. In total, the object tracking pipeline identified 303 918 objects.The histograms of their areas and lifetimes are given in Fig. 5.It would seem that small objects clearly prevail, which alsousually have a short lifetime. There is a peak in the histogramsof areas of the detected objects around an area of 2 Mm , which could be considered as the typical object size. This value corre-sponds to roughly 15 px , thus this is not noise. The inset his-togram with the bin size of 0.267 Mm , which is twice as largeas the data sampling, shows the details around the peak. Thelifetime, on the other hand, seems to continuously decrease from Article number, page 6 of 14vanda et al.: Evolution of magnetic fragments in active regions ]0.02.55.07.510.012.515.017.520.0 O b j e c t li f e t i m e [ h o u r s ] Correlation coefficient 0.31 l o g ( F r e q u e n c y [ n o r m a li s e d ]) Fig. 6.
2D histogram of object area and lifetime. The histogram is plot-ted in the logarithmic scale. It indicates that there is no significant cor-relation between the size and the lifetime of the objects. about 24 minutes (2 frames at a cadence of 12 min per frame) tolarger values.The area and the lifetime of the objects only correlate weaklywith a correlation coe ffi cient of about 0.3. The density histogramof this dependence is given in Fig. 6. We note that the colourscale is plotted in a logarithmic scale; otherwise, the plot wouldbe strongly dominated by a peak in the lower-left corner.The parameters of the individual fragments are strongly af-fected by our methodology, which is especially true for their life-times. During the splitting event, there are more choices as tohow to select the successor of the mother object. Our choice ofthe darker (there we can assume a stronger magnetic field) frag-ment being the successor performs well, except for the type ofevent when a small dark object splits from an evolved sunspot.An area averaged intensity of the sunspot with a penumbra mightbe larger (due to the relative bright large-area penumbra) than thearea-averaged intensity of the small fragment, hence the smallfragment keeps the label of the mother object. In the extremecase of the sunspot with an extended penumbra, it may changethe identity due to splitting frame by frame. Fortunately, in oursample, this does not occur too often overall. After removing ob-jects with a lifetime shorter than 1 hour and a size larger than 706Mm (this roughly corresponds to a size of a supergranule), thecorrelation coe ffi cient between the area and the lifetime is about0.5. The consecutive analysis is not a ff ected by this bias in thelifetime of the objects. According to the scenario by Garcia de La Rosa (1987), thesunspot is formed by merging events from the fragments. Thatwould indicate that the distance between the merging events andthe real sunspots should be rather small. Thus we investigatedthe histogram of distances between the positions of the mergingevents and the closest sunspot separately for forming and de-caying active regions. This is displayed in Fig. 7a and Fig. 7b.Obviously, the merging events occur most often next to the clos-est sunspot, thus supporting the scenario by Garcia de La Rosa(1987). There seems to be a di ff erent distribution function forthis quantity for the forming and decaying regions, where the width of the distribution in distances seems to be larger in thecase of the decaying regions. This indicates that there are ran-dom interactions between fragments occurring away from thesunspots during the active regions’ decay.The same holds true for the location of the splitting events(see Fig. 7c,d). This would indicate that there is a slight prefer-ence of the merging and splitting events to occur very close to thesunspot location in the case of the forming active region, whereaswhen the active region is in the decaying phase, the connectionis somewhat loose. Local helioseismology showed that there arelarge-scale inflows around active regions that are mostly presentduring the later emerging and stationary phases and they seemto weaken during the decaying phase (Haber et al. 2001; Kommet al. 2007; Švanda et al. 2008). The existence of these inflowsduring the formation phases and their lack during the decayingphase would explain our observation. Also, a wider distributionof distances counts the possibility of the emergence of the sec-ondary polarities, which usually does not happen in the earlyphases of the active region evolution.Only with the aim of complementing the picture, we plottedhistograms of distances for the gravity centres of the clusters ofmerging and splitting events from the closest sunspot in Fig. 7e,f.Again, there is a clear preference for the correspondence of thepositions of the event clusters with the location of sunspots. Notsurprisingly, the connection is again somewhat loose in the caseof the decaying active regions.The histogram of distances indicates that most of the merg-ing or splitting events occur near the evolved sunspots. This isfurther strengthened by analysis of the area ratios of the twofragments entering the merging event or splitting into two. His-tograms of these, again, separately for active regions under for-mation and active regions under decay, are plotted in Fig. 8. Onecan clearly see that in all cases, the area ratios are very smallin general, that is, preferably, a small fragment merges with alarger one, or the large fragment splits into two, where one ismuch larger than the other. The latter process resembles an ero-sion rather than splitting.It can also be seen that the smaller area ratios are slightlypreferred in the case of the forming active regions, whereas thedistribution seems wider in the case of the decaying active re-gions. This would indicate that during the formation, the merg-ing of the fragments with the existing sunspots is indeed pre-ferred. Whereas during the decay, the merging of the newly ap-pearing fragments somewhere in the field of view is also morelikely. In the decaying active regions, the number of area ra-tios larger than 0.5 (objects undergoing the merging or splittingevents have comparable sizes) is larger by about 20% than in thecase of forming active regions. As prevously pointed out, for each investigated active region,we obtained a large set of descriptive parameters. They are bestdescribed by their evolution in time, as it is possible to see inFig. 4 for an example. We consider some of these quantities im-portant, which included the number of detected sunspots and thetotal object areas. We also considered the instantaneous balancebetween the splitting (positive) and merging (negative) events.This quantity shows whether (at the given time) merging or split-ting prevails. Furthermore, we evaluated the balance betweenthe newly appearing objects (not by splitting) and faded ob-jects (not by merging). This quantity shows whether, at the giventime, the emergence of the objects prevails (the positive value)
Article number, page 7 of 14 & A proofs: manuscript no. 40127
Forming ARs Decaying ARsa) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Merger to spots distances b) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Merger to spots distances c) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Splitter to spots distances d) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Splitter to spots distances e) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Cluster to spots distances f) F r e q u e n c y [ n o r m a li s e d ] binsize = 1 Mm0.0% outliers Cluster to spots distances
Fig. 7.
Histograms of distances between the positions of the merging events to the closest sunspot (top), position of the splitting events to theclosest sunspot (middle), and between the gravity centres of the merging and splitting clusters and the closest sunspot (bottom). The histogramsare plotted for the active regions in their emerging and formation stage (left) and for the decaying active regions (right). It is important to take noteof the logarithmic scales on the vertical axes. or whether their disappearance prevails (the negative value). Fi-nally, we recorded the total number of the detected objects at agiven time.For every active region in the sample, it is possible to in-vestigate the mutual relations by evaluating the Spearman’s rankcorrelation coe ffi cient; however, we tested to see if the use of thePearson’s correlation coe ffi cient essentially provides the same results. We were looking to investigate whether the behaviour issomewhat common to all active regions. Thus we investigatedthe statistics (by histograms) of the correlation coe ffi cients.The histograms of the correlations of relevant quantities aregiven in Fig. 9. They are plotted in two versions: The blueish barsaggregate all the correlation coe ffi cients that were computed sep-arately for each active region in the sample and the orangish bars Article number, page 8 of 14vanda et al.: Evolution of magnetic fragments in active regions
Forming ARs Decaying ARsa) F r e q u e n c y [ n o r m a li s e d ] b) F r e q u e n c y [ n o r m a li s e d ] c) F r e q u e n c y [ n o r m a li s e d ] d) F r e q u e n c y [ n o r m a li s e d ] Fig. 8.
Histograms of area ratios of fragments during the merging (top) and splitting (bottom) events for the set of forming (left) and decaying(right) active regions. only aggregate those correlation coe ffi cients, whose p -valueswere smaller than the significance threshold (0.05). The p -valuegives the probability that the correlation indicated by the cor-relation coe ffi cient is only apparent and the two correlated se-ries are in fact independent. Hence, in the case when the p -valuewas larger than the threshold, the null testing hypothesis of non-correlation of the series was taken into account and the value ofthe correlation coe ffi cient was thus set to 0.0 for the given pair.As one can see, the di ff erences between the two approaches arenot significant. The histograms were normalised to the area sothat the area integral (or sum strictly speaking) of the histogramsover all bins yields unity.The displayed correlations seem to behave similarly for bothsets of forming and decaying active regions with the exceptionof the correlation of the quantities with the number of sunspots.In the case of the forming active regions, positive or slightly pos-itive correlations prevail; while in the case of the decaying activeregions, the correlations group around zero.For instance, the relation between the number of sunspotsand the total object area is clearly shifted towards large posi-tive correlations (Fig. 9 a). During the emergence, new sunspotsappear and grow in size. The growing objects are classified assunspots eventually. On the other hand, for the decaying ac-tive regions, the correlation takes all values with a strong peakaround zero (Fig. 9 b). This could indicate that during the de-cay, sunspots likely fragment to smaller pieces. Their total area decreases as well but on a longer timescale than the typical split-ting timescale. Hence the correlation between the sunspot num-ber changes, which indicates the evolutionary state of the ac-tive regions, and the change of the total area shows a large peakaround zero and a broad distribution in the values.The cumulative curve of the splitting and merging events inthe case of the forming active regions is dominated by mergingevents (see Fig. 4 upper right panel). The prevailing rather pos-itive correlation between the instantaneous splitting and merg-ing curve and the number of sunspots (Fig. 9 c) is dominatedby sunspot merging in the forming phase of the active region.Many fragments in this phase have already been classified assunspots (see a further discussion of Fig. 9 g in the next para-graph) and thus the number of merging events correlates with adecrease in the number of sunspots. This is consistent with thework by Kutsenko et al. (2019), where they point out that duringthe emergence, the coalescence is an important process; how-ever, the emergence of large structures brings more energy to theforming active region. In the case of the decaying active regions,the correlation between the same quantities is zero (Fig. 9 d).The splitting and merging curve is dominated by splitting eventsin this case. However, the number of sunspots does not increaseaccordingly. This may be understood in a way that the splittingevents result in small objects that are not classified as propersunspots. This again resembles an erosion process rather thansplitting into two comparable fragments. Article number, page 9 of 14 & A proofs: manuscript no. 40127
Forming ARs Decaying ARsa) F r e q u e n c y [ n o r m a li s e d ] vs . Total object area [Mm ] b) F r e q u e n c y [ n o r m a li s e d ] vs . Total object area [Mm ] c) F r e q u e n c y [ n o r m a li s e d ] vs . Splitting (+) / merging (-) d) F r e q u e n c y [ n o r m a li s e d ] vs . Splitting (+) / merging (-) e) F r e q u e n c y [ n o r m a li s e d ] vs . f) F r e q u e n c y [ n o r m a li s e d ] vs . Fig. 9.
Histograms of correlation coe ffi cients between various quantities describing the evolution of the fragments in the forming (left) and decaying(right) active regions. The blue bars represent the the histograms of all correlation coe ffi cients, whereas the orange-bar histograms were derivedusing only the statistically significant correlations coe ffi cients. The brown colour indicates the overlap of both types of histograms. The number of sunspots and the curve of newly appearing(not by splitting) and fading (not by merging) objects (Fig. 9 e,f)are positively correlated in the case of forming active regions.This is consistent with the emergence of new objects dominatingtheir spontaneous disappearance during the emerging phase ofthe active region when the number of identified sunspots growsin general. In the case of decaying active regions, the imbalancebetween the newly appearing and fading objects has nothing to do with the actual number of remaining sunspots in the area.This is consistent with a random process, which is further sup-ported by the distribution of distances between the merging andsplitting events and the sunspots. This distribution of distances iswider in the case of decaying active regions (Section 4.3). If theactive region in the decaying phase is already disconnected fromits magnetic roots (suggested e.g. by Fan et al. 1994; Schüssler& Rempel 2005), the emergence of the new dark objects must
Article number, page 10 of 14vanda et al.: Evolution of magnetic fragments in active regions
Forming ARs Decaying ARsg) F r e q u e n c y [ n o r m a li s e d ] vs . h) F r e q u e n c y [ n o r m a li s e d ] vs . i) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . j) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . k) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . l) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . Fig. 9.
Continued be due to the dynamo process operating in the near-surface lay-ers. A similar observation is drawn about the correlation betweenthe number of the detected sunspots and the number of objects(Fig. 9 g,h). The weak positive correlation for the forming activeregions has two interpretations. Either a considerable fraction ofobjects already emerge as objects that are large enough to beclassified as sunspots, or these objects emerge with the proper-ties below the sunspot classification but coalesce quickly withthe other objects and become sunspots. We remind our readersthat the cadence of our dataset is 12 minutes. Both interpreta- tions are consistent with the observation that often, when an ob-ject is registered for the first time, it already has the propertiesof sunspots. To shed some light on the discrimination betweenthese two hypotheses, one would need to study the datasets witha higher temporal resolution, for instance with the origin 45-ssampling of the HMI. Unfortunately, such a test is currently be-yond our computer capabilities. In the case of decaying activeregions, the number of objects has nothing to do with the num-ber of detected sunspots.
Article number, page 11 of 14 & A proofs: manuscript no. 40127
Forming ARs Decaying ARsm) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . Splitting (+) / merging (-) n) F r e q u e n c y [ n o r m a li s e d ] Total object area [Mm ] vs . Splitting (+) / merging (-) o) F r e q u e n c y [ n o r m a li s e d ] Splitting (+) / merging (-) vs . p) F r e q u e n c y [ n o r m a li s e d ] Splitting (+) / merging (-) vs . q) F r e q u e n c y [ n o r m a li s e d ] vs . r) F r e q u e n c y [ n o r m a li s e d ] vs . Fig. 9.
Continued
The total area covered by the objects weakly correlates withthe total number of identified objects (Fig. 9 i,j). The distribu-tion of the correlation coe ffi cients is similar for both the formingand decaying active regions; however, it is slightly shifted to-wards larger values in the case of the forming active regions.This weak correlation indicates that the sizes of the objects dif-fer and no typical size exists. If a typical size existed, we wouldexpect a clear (even linear) correlation between these two val-ues. The same conclusion may be drawn about the correlation between the total area and the imbalance between the newly ap-pearing and spontaneously disappearing objects (Fig. 9 k,l).Something that is somewhat surprising is a weak positivecorrelation between the total area covered by the objects and theimbalance between the number of splitting and merging events(Fig. 9 m,n). One would naively expect that during the splittingand merging, the area is conserved. Thus when the splitting oc-curs, the splitting and merging curve increases, but the total areashould stay the same size. When the merging occurs, the split-ting and merging curve decreases, but the total area should re- Article number, page 12 of 14vanda et al.: Evolution of magnetic fragments in active regions F r e q u e n c y [ n o r m a li s e d ] Fig. 10.
Histogram of maximal distances of fragments during its life-time from the position where it merged with another one. It is importantto note the logarithmic scaling on the vertical axis. main constant again. Therefore the expected correlation coe ffi -cient value is zero. The positive values of correlation indicatethat the summary area of the resulting fragments is larger thanthe area of the mother object and, similarly, that for the mergingevent, the merger has a smaller area than the sum of the areas ofthe predecessor objects. If we assume that the total magnetic fluxis conserved during the merging and splitting events, we may saythat the average magnetic induction of the objects decreases aftersplitting and increases after merging.Unfortunately, a finite spatial resolution may play a signifi-cant role here. The HMI resolution will likely lead to an over-estimation of the areas of smaller fragments, against the area ofbigger fragments. The bias will be relatively larger for very smallfragments, as the accuracy of measuring the area is higher whenthe area is significantly larger than the pixel size. A coarse gridof pixels may lead to a relatively lower intensity of small struc-tures, as a result of smaller filling factor, but to an overall largerarea, unless the contrast is low enough to inhibit its detection.Therefore the positive correlations seen in Fig. 9 m,n may be aconsequence of the sampling.The splitting and merging and newly appearing and spon-taneously disappearing curves weakly correlate with the totalnumber of objects (Fig. 9 o,p and q,r). This indicates that nei-ther of the processes is dominant in any phase of the active re-gion lifetime. The correlation seems a bit larger for the splittingand merging curve, showing slightly greater importance of thisprocess on the total number of detected objects. We now focus on merging events, namely on the evolution of thefragment between its appearance and its merging with anotherfragment. We have shown that in most cases, it is the smallerfragments that merge with the larger ones. Their lifetime is rathersmall, a few hours at most, but most often it is less than one hour(i.e. five frames taking the cadence of our data into account).We studied the statistics of the maximum distance during theobject lifetime and the position of its death by merging. Fig. 10shows that in most cases, this distance is rather small with a max-imum at about 2 Mm.We also investigated, how fast do the fragments approacheach other. Fig. 11 indicates that these speeds are rather slow, F r e q u e n c y [ n o r m a li s e d ] Fig. 11.
Histogram of speeds of the fragments approaching the mergingevent. We note the logarithmic scaling on the vertical axis. F r e q u e n c y [ n o r m a li s e d ] Fig. 12.
Histogram of speed standard deviations expressed as the frac-tion of the mean speed of the fragments approaching the merging event. about 0.5 km / s, which also corresponds to the minimum veloc-ity detectable given the cadence and sampling of the data. Wehave to add that null velocities were removed from this plot. Thespeed of 0.5 km / s corresponds to the typical speed of the super-granular di ff usion. In studies performed using high-resolutionobservations, such as Guglielmino et al. (2010) for example,higher speeds were observed for small magnetic elements, wherethe recorded motions were dominated by granular flows, whichmay be an order of magnitude larger. Such small scales are outof reach when using HMI / SDO measurements.We studied, how does this velocity vary during the motion.Fig. 12 indicates that the speed does not vary much as the dis-tribution of the speed standard deviations that are normalised bythe mean speeds is a heavy-head distribution. This indicates thatthe approaching speed towards the merger of the fragments ismore or less constant during the whole process.We were interested to see, whether the fragments go straighttowards the merging centre or whether this process is rather sim-ilar to the random-walk process. The instant directions towardsthe merger positions are defined by the angle ϕ ascos ϕ = v · m | v || m | , (2) Article number, page 13 of 14 & A proofs: manuscript no. 40127 F r e q u e n c y [ n o r m a li s e d ] Fig. 13.
Histogram of the fragment’s directions towards the mergingcentres. where v is an instant vector of the fragment velocity and m is thevector towards the position of the final merging event, definedby m = r m − r . (3)Here, r is an instantaneous position of the fragment and r m is theposition of the final merging event.The histogram of ϕ displayed in Fig. 13 indicates that, ingeneral, fragments steer towards the merging position, but notquite straight. The variations are large, similar to a random-walkprocess. The peaks at multiples of 90 degrees are a consequenceof the data sampling; a better spatial resolution would likely re-move these peaks.
5. Conclusions
We studied the critical evolution phases of more than 1000 ac-tive regions observed by the HMI / SDO instrument. The sam-ple contained active regions both undergoing formation and de-caying. Using a feature-recognition and tracking code, we au-tomatically detected dark features representing magnetised fea-tures from small elements to proper sunspots. We tracked theseobjects through the spatio-temporal datacube and followed theirevolution. We recorded events when these objects merged withother object and events when the object split in two. We studiedthe statistics of these events.The lifetime of the detected objects is rather short, typicallya few tens of minutes. For large objects, the determination of thelifetime may be biased by our methodology. They merge withother objects or fade soon after their appearance. Their typicalsize seems to be around 2 Mm . Due to the short lifetime, thenumber of the appearing (by the emergence of splitting) objectsand disappearing (by fading or merging) objects is in a surprisingbalance throughout the typical lifetime of the active region.We obtained our result by automatic processing of the largesample of active regions. Statistically, we confirm that sunspotsform by merging events of smaller fragments. The coalescenceprocess is driven by the turbulent di ff usion in a process similarto random-walk, where supergranular flows seem to play an im-portant role. The number of the appearing fragments does notseem to significantly correlate with the number of sunspots thatformed. The formation seems to be consistent with the magneticfield accumulation. Statistically, the merging occurs between a large and a much smaller object. The decay of the active regionseems to take place preferably by a process similar to the ero-sion.Both merging and splitting events preferably occur in the im-mediate vicinity of evolved sunspots. Merging or splitting fur-ther away is rare. The merging and splitting events do not occurat random places, the preferential locations cluster again in aclose vicinity to the sunspots.Using HMI / SDO intensitygrams, we are unable to assess thescenario by Garcia de La Rosa (1987). After a fragment mergesinto a sunspot, the track of its location and evolution is lostmainly due to the small spatial resolution of 1". With HMI / SDOintensitygrams, it is not possible to identify and track the darknuclei within sunspot umbrae, which are seen in high-resolutionobservations (e.g. Sobotka & Puschmann 2007). These dark nu-clei probably correspond to the surviving fragments indicatedby the scenario by Garcia de La Rosa. Higher resolution obser-vations perhaps with a better dynamical range are needed. Thisissue thus remains open.
Acknowledgements.
The authors were supported by the Czech Science Foun-dation under the grant project 18-06319S. ASU CAS is funded by the instituteresearch project ASU:67985815. We thank the referee for helpful suggestionsfor improvements of this paper. The SDO / HMI data are available by courtesy ofNASA / SDO and the HMI science team.
References