Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
B. Lemmerer, D. Utz, A. Hanslmeier, A. Veronig, S. Thonhofer, H. Grimm-Strele, R. Kariyappa
AAstronomy & Astrophysics manuscript no. LemmererBirgit c (cid:13)
ESO 2019May 10, 2019
Two-dimensional segmentation of small convective patterns inradiation hydrodynamics simulations
B. Lemmerer , D. Utz , , A. Hanslmeier , A. Veronig , S. Thonhofer , H. Grimm-Strele , and R. Kariyappa Institute of Physics, IGAM, University of Graz, Universit¨atsplatz 5, 8010 Graz, Austria Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Apdo. de Correos 3004, 18080 Granada, Spain Institute of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria Indian Institute of Astrophysics, Koramangala, 560034 Bangalore, IndiaReceived: 29 March 2013 / Accepted: 6 February 2014
ABSTRACT
Context.
Recent results from high-resolution solar granulation observations indicate the existence of a population of small granularcells that are smaller than 600 km in diameter. These small convective cells strongly contribute to the total area of granules and arelocated in the intergranular lanes, where they form clusters and chains.
Aims.
We study high-resolution radiation hydrodynamics simulations of the upper convection zone and photosphere to detect smallgranular cells, define their spatial alignment, and analyze their physical properties.
Methods.
We developed an automated image-segmentation algorithm specifically adapted to high-resolution simulations to identifygranules. The resulting segmentation masks were applied to physical quantities, such as intensity and vertical velocity profiles, pro-vided by the simulation. A new clustering algorithm was developed to study the alignment of small granular cells.
Results.
Small granules make a distinct contribution to the total area of granules and form clusters of chain-like alignments. Thesimulation profiles demonstrate a di ff erent nature for small granular cells because they exhibit on average lower intensities, lowerhorizontal velocities, and are located deeper inside of convective layers than regular granules. Their intensity distribution deviatesfrom a normal distribution as known for larger granules, and follows a Weibull distribution. Key words.
SUN: granulation- convection - Techniques: image processing
1. Introduction
New high-resolution solar telescopes provide us with theopportunity to study the solar photosphere in more detailthan ever before. The recent achievements in high-resolutionobservations ( ∼ . data regarding the analysis of the solar convection is that wedo not need to consider the influence of magnetic fields onthe convective pattern. For the exclusive investigation of thenewly discovered mini-granules, bright grains originating fromsmall-scale magnetic fields would be particularly interfering.Therefore, fields such as magnetic bright points (MBPs, seee.g. de Wijn et al. 2009; Utz et al. 2010) would have to beexcluded from the analysis, but they inherently not present inRHD simulations.For the purpose of detecting small-scale convective patterns,a new segmentation algorithm was developed. It takes severalphysical properties of the convection into account that are pro-vided by the simulation, such as the vertical velocity and the More details on the state-of-the-art MHD simulations can be foundin Nordlund et al. (2009). 1 a r X i v : . [ a s t r o - ph . S R ] M a y . Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations Fig. 1.
Snapshot of the simulated data computed by the ANTARES code during a 3D RHD run. In this model, the temperaturedata are color coded. Moreover, a temperature iso-surface of 6000 K, located at a height of ∼
750 km below the upper boundary,indicates the granular pattern. An artificial light source was placed above the model to accentuate the structure of iso-surface.emerging intensity which is defined as the outwardly directedintensity that leaves the computational box through the upperboundary. Furthermore, a clustering algorithm was developedand applied to the segmented data to study the alignment of smallgranules.
2. Simulation setup and data
ANTARES (a numerical Tool for astrophysical research)is an RHD code for numerically simulating the solar near-surface convection (box-in-a-star approach) developed byMuthsam et al. (2007, 2010). The FORTRAN90-code solvesthe set of RHD equations using weighted essentially non-oscillatory (WENO) high-resolution numerical schemes (seealso Zaussinger & Spruit 2013; Mundprecht et al. 2012).Open boundary conditions in the vertical direction allowfree in- and outflow, while in the horizontal directions periodicboundary conditions are used. A detailed description of theboundary conditions can be found in Grimm-Strele et al. (2013).In the upper ∼
30% of the simulation box the radiative heatingrate is calculated using gray approximation, whereas in the restof the box di ff usion approximation is valid. The 3D model isinitialized from a 1D model to which a weak perturbation isadded to break the horizontal symmetries. The simulation isthen thermally relaxed in an almost two-hour long period, whichprovides su ffi cient time for the development of 3D structures.The box of the 3D model comprises 9 Mm in horizontaldirections and 5 Mm in vertical direction. The spatial resolutionin the horizontal directions is 32.1 km, in vertical direction15.3 km, resulting in 281 by 339 grid cells. To study thegranular cells we processed a data set with a temporal resolutionof ∆ t =
30 s. Convective patterns of the simulated two hours ofreal solar evolution were analyzed at the calculated bottom ofthe photosphere.Figure 1 illustrates the temperature distribution of a 3D snap-shot from an ANTARES model run. The granular pattern in thegiven temperature iso-surface at 6000 K is clearly visible.
3. Automated granule detection in two dimensions
To analyze of the numeric RHD data we developed a two-dimensional segmentation algorithm. The automated detectionof granular cells is crucial for processing large datasets andthe successive statistical analysis. Several sophisticated algo-rithms to segment the solar granulation in observational dataexist and are publicly available. Among them is the so-calledmultiple-level tracking (MLT) algorithm developed by Bovelet& Wiehr (2001), which was previously tested on ANTARESdata (see Lemmerer et al. 2012). Our algorithm is based on thesame basic idea of using multiple thresholds to segment imagedata, but it is also di ff erent in several crucial parts and details.The incorporation of methods of pattern recognition, such asedge-detection routines and morphological operations as wellas of velocity maps of the surface flows (dopplergrams) resultedin a fast and reliable segmentation routine, which is not onlyuseful for high-resolution data from observations, but also forthose obtained from simulations. In addition, we developed atool analyzing the clustering behavior of small granular cells.To segment the granular structures in two dimensions, a ref-erence level within the segmentation box has to be determined.To do this, we calculated the geometric height (where the opticaldepth τ =
1) to define the bottom of the photosphere . We eval-uated the equation of optical depth for each grid point columnby column from model data values. Then we extracted for eachpoint the height at which the optical depth reaches unity. Fromthis we were able to construct the surface of optical depth unity,which we refer to as τ -iso-surface (see also Leitner et al. 2009;Lemmerer et al. 2010). The two-dimensional segmentation algorithm was applied tothe afore mentioned τ -iso-surface. The algorithm di ff ers insome aspects from segmentation algorithms that are based Because we used the gray approximation, the wavelength is irrele-vant.2. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 2.
Steps of the applied segmentation: (a) normalized emerging intensity and (b) vertical velocity on the τ -iso-surface of theANTARES simulation, (c) initial vertical velocity threshold applied to the vertical velocity on the τ -iso-surface, (d) subsequentlower threshold applied to the structure, (e) lowest threshold applied to the structure, (f) edge detection applied to the emergingintensity of the object, (g) granular cells still connected by pixels after applying of the lowest threshold, (h) morphological openingto separate connected granular cells, (i) final segmentation mask.solely on intensity observations (filtergrams). Data recordedwith high-resolution telescopes provide in most cases only in-tensity information in the form of gray-level images. Therefore,algorithms are often based on thresholding of multiple intensitylevels (e.g. MLT) or on image-processing routines such asskeletonization (e.g. Florio & Berrilli 1998).The algorithm introduced here is based on a multiple threshold-level segmentation and on image-processing techniques. Thesegmentation additionally profits from the use of physical quan-tities, that is the combination of vertical velocity and intensity or temperature that are o ff ered by the ANTARES model. Thewhole scheme is a bottom-up approach. Large fragments aresequentially broken down into smaller structures by changingthreshold levels, that is, changing them to higher velocities,which results in smaller structures . The opposite scheme, a top-down approach, is used in the MLTalgorithm (see Bovelet & Wiehr 2001). Starting from high-intensityseeds, the granular segments grow by shifting the threshold level tolower intensity values. 3. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
The algorithm starts with the pre-processing of the verticalvelocity (Fig. 2b) and the emerging intensity (Fig. 2a). Thisincludes normalizing both surface profiles to the interval [0 , . While the algorithm can beapplied to intensity as well as to vertical velocity profiles,for the current study we used the vertical velocity on the τ -iso-surfaces (see Fig. 2c) as input for the thresholding. Theintensity profiles are required for later image-processing opera-tions. The resulting structures are more distinctly separated thanthose obtained from the intensity segmentation. If vertical veloc-ities are not available, intensity images can be processed instead.After applying the initial threshold on the velocity on the τ -iso-surface, the resulting binary image is labeled and each seg-mented structure is analyzed. After each thresholding step, theinitial threshold is again reduced by a factor that is another in-put parameter. Next, this thresholding routine is applied to eachof the previously found and labeled structures (see Fig. 2d). Ifthe previously found structure is separated because of the cur-rent thresholding, new labels are applied to the found structures.This scheme is repeated on these new structures. The thresh-olding routine is iterated until the lowest threshold is success-fully applied (see Fig. 2e), that is until the strongest upflows andsmallest (most fragmented) structures are detected or as long asthe segmented objects meet the following conditions: – the diameter of the structure exceeds the largest diameter ofgranules , – The structure falls below a critical value of its solidity , in-dicating that granular objects are still connected.After the final threshold was applied and objects meet the pre-vious conditions, morphological operations are executed in thefollowing way: – Laplacian-of-Gaussian edge-detection is applied to the pro-files of the emerging intensity of the structure to detectclosed contours (Gonzalez et al. 2009), see Fig. 2f, – a morphological opening is performed to separate objectsconnected by a few pixels, as illustrated in Fig. 2g, as well asto preserve the shape of the objects (Gonzalez et al. 2009),see Fig. 2h.Applying the edge-detection routine to the emergingintensity profits from the high-intensity contrast between inter-granular lanes and granular cells, which leads to a well-definedseparation of merged structures. The high-intensity contrastis exploited in cases of merging of granules. When the initialphase of merging occurs in an analyzed velocity profile, theemerging intensity still shows a dark intergranular lane. These Because the intensity distribution varies during the whole time se-ries, the lowest and highest segmentation thresholds are shifted by therespective change in the mean image intensity. In particular, this shiftingtakes care of the intensity variations caused by the 5-minute oscillation. The diameter of a granular cell is defined as the equivalent diameterof a circle with the same area as the region, computed via √ ∗ A /π . The largest diameter was derived from the probability density func-tion of the granule diameters (see section 4). The solidity of an object is defined as the ratio of its area to thearea enclosed by its convex hull. A solidity of 0.7 is used as the criticalvalue. lanes indicate clear separations between granular cells. On theother hand, these granules would already appear as a mergedstructure in the vertical velocity on the τ - iso- surface. Hence,additionally using the emerging intensity is of advantage for acorrect segmentation.Furthermore, incorporating the emerging intensity helps dur-ing the initial phase of the fragmentation process of explodinggranules . These features often show a concavity at their centercaused by downflowing plasma. In the emerging intensity thesedownflows appear as a depression on or outside of the granule.To prevent a loss of information regarding the statistics andphysical properties of these granules, holes are automaticallyfilled if the plasma downflow is located at the center of thegranule. The structure is split into several substructures if thedownflow causes a fragmentation.The periodic boundaries are taken into account by excludinggranular fragments, separated by the right and lower boundary(see Fig. 2i), and appending them to the left and upper bound-ary, respectively. Later, this allows a calculation of the correctareas, diameters, and centroids of the granule split by the peri-odic model boundaries. The applied image-processing steps anda labeling of all objects result in the final segmentation mask (seeFig. 2i). The border handling of split granules is illustrated bythe assigned labels depicted as gray values. The mask is then ap-plied to various profiles of physical quantities to extract statisti-cal information of the identified granules. The final step retrievesinformation such as granular area, perimeter, mean intensities etcetera. According to Abramenko et al. (2012), mini-granules thatform clusters and chains are situated in the intergranular lanes.To investigate this observational finding in the ANTARESsimulation and to quantify the finding in a statistical analysis,we developed a clustering algorithm adapted to the segmenteddata.As described in section 3, at each time step a segmentationmask is produced. The clustering algorithm is based on thesefinal segmentation masks. The masks are labeled and statisticalinformation, such as the area and the centroid of each granulesis derived. The clustering algorithm starts with the detectionof small granules with diameters smaller than the thresholddiameter of about 750 km .These small granules are labeled in ascending order and sep-arated from the remaining larger granules. Each of them (shownin red in Fig. 3a) is examined separately. Euclidean distancesbetween the centroids (shown in pink in Fig. 3b) of the inves-tigated granular cells and all other granules are calculated. Thefour nearest granules (plotted in black in Fig. 3b) with a mini- Exploding granules are the result of the final phase of huge granulesstarting with a ceasing upflow in the middle of the granules. The still-expanding plasma plus the radiative cooling in the central region finallyleads to a flow reversal in the middle of the granule with a downflowsetting in. This downflow either creates a depression in the middle ofthe granule or leads to its fragmentation. This diameter was determined by the global maximum in the prob-ability density function of the diameters of granules (see section 4).4. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 3.
Steps of the clustering algorithm: (a) segmented image including identified small granular cells shown in red, (b) calculationof Euclidean distances to neighboring granules, (c) final result of the clustering illustrated in di ff erent colors.mum distance to the examined one (red) are now analyzed (cor-responding Euclidean distances are listed in Fig. 3b). Severalcriteria define the assignment of the investigated granular cellsto a cluster: – if the examined granular cell is surrounded only by largecells and the nearest small cell is by definition located toofar away (distance larger than 1.9 Mm), which is the case inFig. 3b, the examined cell is classified as not belonging to acluster (Fig. 3c), – if the examined granule has another small granule within itsclosest four neighbors, which does not already belong to acluster, this neighboring small granule adopts the pixel value(label) of the examined granule, – if a neighboring small cell already belongs to a cluster, theexamined cell receives the same label (pixel value) and isappended to the cluster.The algorithm results in an image consisting of labeled clusters,where each cluster itself consists again of several isolated smallgranules uniquely belonging to one of those clusters. Figure 3cindicates the clusters in di ff erent colors. Statistical properties,such as centroids and eccentricities of the clusters, are deter-mined. The possibility to allocate clusters formed by severalmini-granules enables us to quantify their spatial distributionin the field of view. The determination of the eccentricity ofthe clusters is used to determine the alignment of small granuleswithin these clusters.
4. Results
The developed two-dimensional segmentation algorithm o ff ersthe possibility to retrieve statistical information of the seg-mented granular cells. For our analysis we used a two-hour dataset consisting of 281 time steps with a temporal resolution of30 seconds. Each time step contains an average of 93 granules,yielding a total statistical sample of about 20 500 granules.Physical properties, such as the vertical and horizontal velocityas well as the geometrical depth of the segmented granules are The eccentricity of a cluster is defined as the ratio of the distancebetween the foci of the ellipse enclosing the cluster and the length ofthe major axis. The value of the eccentricity is between 0 and 1. evaluated on surfaces of the optical depth τ =
1. The emergingintensity of the segmented granules is analyzed at the top ofthe computational box. This can be understood as the intensitythat reaches a virtual observer’s telescope. At each time instantthe segmentation mask is applied to the emerging intensity andto the profiles of the vertical and the horizontal velocities, toobtain characteristic parameters for granules, such as meanintensity, geometrical depth, and horizontal and vertical velocityvalues. These physical properties of identified granules are thenanalyzed with respect to their equivalent diameters via scatterplots.Figure 4a shows the segmentation mask colored accordingto the vertical position of the surface of optical depth unity.We can see that small granules are on average located deeperthan the larger ones, which is also apparent in the scatterplot in Fig. 4b. The ordinate in Fig. 4b indicates the meangeometrical depth of the detected granules in the simulationbox in km. Negative values indicate the location of a granulewithin the convective zone. The scatter is distributed in twodistinctive regions with di ff erent trends. Thus, the applied twolinear fits separate the scatter for the two distinctive regimes.The two linear fits intersect at a diameter of ∼ ∼
5. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 4. (a) Application of the segmentation mask to the geometrical depth of the iso-surface of τ = τ = τ = Fig. 5. (a) Application of the segmentation mask to the normalized emerging intensity for a single frame. (b) Granular diameter vs.normalized mean emerging intensity of granules for the entire time series. The trend of the scatter is illustrated by linear fits (solidblack lines) and a polynomial fit (dashed dark-gray line).distributions di ff er from Gaussians but fit a Weibull distribution,as illustrated by the solid red and blue lines in Fig. 6a. Asa consequence, we regrouped granules according to theirdiameters in several subpopulations and analyzed their intensitydistributions separately. In this way, we obtained a family ofWeibull curves with di ff erent shape parameters (see Fig. 6b).We plot these parameters versus the group diameters in Fig. 6c.The figure demonstrates that the best-fitting Weibull distributionshows a linear parameter dependence on the mean diameter ofgranules. The theoretical interpretation of the shape parameter,its dependence on granular diameters and the meaning of theWeibull distribution for the physics of the solar granulation hasto be investigated in more detail and will be a possible topic forfuture study.We also analyzed the mean vertical velocity of segmentedgranules (Fig. 7a). The distribution of the mean vertical velocityof identified small granules exhibit a large scatter, which may be related to di ff erent stages of their evolution because theyare predominantly found in deeper layers (see Fig. 7b). Thepolynomial and linear fits show a strong decrease of verticalvelocity (upward moving plasma) with decreasing granulardiameters and thus support the concept of the existence of twodistinct populations.Applying the segmentation masks on the horizontal veloc-ity profiles of the RHD model (see Fig. 8a) enables us to esti-mate the mean horizontal velocities (Fig. 8b, red), which reveala slightly decreasing trend over the whole range of granular di-ameters (solid black line in Fig. 8b). We then excluded gran-ules with diameters smaller than 750 km, because they exhibit alarge scatter, and found for the remaining majority of granules apractically constant behavior, illustrated by the dashed approx-imately horizontal dark-gray fit. The maximum horizontal ve-locity (Fig. 8b, blue) is dependent on the granular size. As thediameter increases, the maximum horizontal velocity increases.
6. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 6.
Analysis of normalized mean emerging intensity: (a) comparison of distributions of the normalized mean emerging intensityof large granules shown in green, medium-sized granules in red, and small granules in blue as well as their Gauss and Weibullfits, respectively, (b) family of Weibull curves derived from regrouping granules according to their diameters in classes, (c) shapeparameter of the Weibull distributions derived from the intensity distribution family in (b).This result coincides with findings in simulations (e.g. Ste ff enet al. 1989). The increase in horizontal velocity to balance thelarger plasma volume that ascends to the surface can be under-stood as a consequence of mass conservation (see e.g. Nordlundet al. 2009). Figure 9 shows the area contribution function defined as thecontribution of a given size of a granule to the total area ofgranules (Roudier & Muller (1986)). The function revealsa local minimum located in an area of ∼ . , whichcorresponds to an equivalent diameter of ∼ ∼ .
43 Mm corresponding to anequivalent diameter of ∼
750 km.The probability density function of the equivalent diameter(blue line in Fig. 9) shows an increase in the number of detected granules towards smaller scales. We found a less distinct changein the slope than Abramenko et al. (2012).For our statistical analysis the distinct global maximum at adiameter of 750 km (Fig. 9) was defined as the threshold valueto discern between the population of small granules and largerones. Interestingly, this value agrees well with the points of in-tersection of the linear fits applied to the vertical velocity andhence might have a real physical meaning.
According to Abramenko et al. (2012), small granules oftenform clusters and chains. Our clustering algorithm enables usto quantify the spatial arrangement of these small granules withdiameters smaller than 750 km. Figure 10 shows the distributionof the number of small granules in the clusters. We see that themajority of clusters consist of two granules. Because apparently
7. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 7. (a) Application of the segmentation mask to the vertical velocity on the τ -iso-surface for a single frame. (b) Scatter plotof the mean vertical velocity of segmented granules vs. the granular diameter for the entire time series. The trend of the scatter isillustrated by two linear fits (solid black lines) and a third-order polynomial fit (dashed dark-gray line). Negative values correspondto upflowing plasma. Fig. 8. (a) Application of the segmentation mask to the horizontal velocity on the τ -iso-surface for a single frame. (b) Scatter plotof granular diameter vs. mean horizontal velocity of segmented granules for the entire time series, shown in red and the maximumhorizontal velocity in blue. The trends for granules across the whole range of diameters are illustrated by linear fits as solid blacklines and for granules with diameters larger than 750 km by a linear fit as a dashed dark-gray line (only shown for the meanhorizontal velocity).two granules can only form lines and chains, we adjusted thedistribution by neglecting all clusters formed by two granulesand only considered clusters consisting of three or more gran-ules. The resulting distribution of the eccentricities of clusters isdisplayed in Fig. 10b. We found that small granules in clusterspredominantly form chains. The main part of clusters feature aneccentricity higher than 0.9, that is close to 1 (alignment along aline; a value towards 0 would represent a circular alignment).
5. Discussion and conclusions
New telescopes and 3D simulations produce huge amounts ofdata, which make it necessary to develop algorithms for auto-mated analysis. High-resolution simulations such as those ob-tained with the RHD simulation code ANTARES provide syn-thetic intensity maps and velocity profiles that can be com- pared with observations. Additionally, the simulation data al-lowed us to infer the characteristic physical quantities on anylayers in the simulation box, which are not accessible to obser-vations. The segmentation algorithm developed and describedhere, makes use of this additional information by combining amultiple-level thresholding routine applied to the vertical veloc-ities with image-processing techniques, that use the emergingintensity profiles at the top of the computational box.
Granular characteristics:
The analysis of the horizontal andvertical velocity, and also all intensity distributions of thesegmented granular cells, yealded di ff erent results for smallgranules than for granules larger than 1000 km. Thus we canapparently speak of two distinct populations of granules. Themost distinct characteristics of small granules are their lower
8. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Fig. 9.
Area contribution function shown in red. The black curve indicates the probability density function of the granular area andthe blue solid curv the equivalent diameter of segmented granules.
Fig. 10.
Analysis of the structural alignment of clusters: (a) histogram of the number of granules forming a cluster. (b) Probabilitydensity of the eccentricity of detected clusters that consist of more than two granules.intensities and maximum horizontal velocities. The fits appliedto the scatter plots of the emerging intensity as a functionof granular size indicate a positive slope for granules withdiameters smaller than ∼ ff erent diameters we deduced a linear relationshipbetween the Weibull shape parameter and mean granular classdiameters. The larger the diameter, the more Gaussian-like theintensity distribution. These results agree in so far with findings in Yu et al. (2011), as they found that small granules do notsatisfy Gaussian distributions, whereas the continuum intensitydistribution of large granules is similar to a Gaussian.The analysis of the vertical velocity as depicted in Fig. 7b il-lustrates a considerable scatter for small granules. Especially thesmallest among them show downflowing motions, while largergranules are in a predominately upflowing state. This was con-firmed by observations (Yu et al. 2011) and by results from simu-lations (Gadun et al. 2000). To study the origin of these motions,detailed analysis of the temporal evolution of granules has to becarried out.
9. Lemmerer et al.: Two-dimensional segmentation of small convective patterns in radiation hydrodynamics simulations
Structural appearance and clustering:
The area contributionfunction (Fig. 9) agrees with findings in previous studies. Wedetermined the global maximum at a diameter of 750 km, whichis smaller than the dominant scale. This might be because of thehigh contrast of the data and the possibility to segment granuleswithout loosing information due to the detectability of thesmallest variations of intensity levels within detected granules.Another finding is that granules with diameters smaller than ∼
750 km contribute significantly to the total area of granules.The determined probability density function of the diam-eters of identified granules (Fig. 9) di ff ers from the results inAbramenko et al. (2012), where the probability density functionshows a decrease from small diameters to diameters of up to600 km. This probably results from over-segmentation becausethey used single thresholds to detect granules. While the authorsstated that they excluded magnetic bright points , some ofthem might still have escaped the exclusion. Hence, these smallgranules may have contributed to their analysis, which might beanother reason for the occurrence of many granules identifiedwith a diameter smaller than 300 km, which cause the steepincrease of their curve toward smaller sizes.The eccentricity distribution derived for the clusters indi-cates that small granules form clusters that show chain-likealignments. This coincides well with observational results inAbramenko et al. (2012). General remarks and outlook:
When studying time series withan interval of 30 seconds as used in this analysis, small granulesappear for only a few time steps and rarely reach the same inten-sity as the neighboring larger granules. This behavior suggeststhat small granules, which are situated in the intergranular lanes,are not evolving to the same geometrical height as larger granu-lar cells. Movies suggest that small granules may not result fromfragmentation of larger granular cells but instead evolve and dis-solve in regions of intergranular lanes, rarely merging with othergranules. For a detailed automated analysis of their evolution ahigher time-cadence is necessary. An increase of the temporalresolution by an order of magnitude is desirable for the con-tinuous investigation of the granular evolution. This will con-sequently lead to a better understanding of the evolution of theplasma upflow, manifested in two dimensions at photosphericlevels, as granules. In the future, a more complete understand-ing of the behavior of small granules will be gained by studyingthe 3D evolution of the upstreaming hot plasma plumes in theconvection zone. For this purpose, we plan to develop a general-ization of the segmentation algorithm that can be applied to threedimensions.
Acknowledgements.
The research work was funded by the Austrian ScienceFund (FWF): P23818 and P20762. D.U. is particularly greatful for the specialsupport given by project J3176 (Spectroscopical and Statistical Investigations onMBPs). The model calculations have been carried out at VSC (project P70068,H. Muthsam). A.H., B.L. and D.U. thank the ¨OAD and ICD for financing ascientific stay at the Indian Institute of Astrophysics. R.K. thanks the ICD and¨OAD for financing a short research stay at the IGAM, Institute of Physics, of theUniversity of Graz. The authors thank the anonymous referee for constructivecomments that helped to improve the segmentation algorithm. Small-scale magnetic flux concentrations that appear bright in fil-tergram observations at scales of ∼
200 km in diameter.
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