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during the last few years , our understanding of the structure , organization and evolution of magnetic fields in the very quiet sun ( the regions outside active regions and the network ) has become increasingly clear . magnetic fields in the quietest areas of the sun are relatively weak and organized at small spatial scales , which yields weak polarization signals that are difficult to observe . until very recently , the general picture of the structure of its magnetism was rather rough : a `` turbulent '' disorganized field @xcite . it is now clear that even in very quiet areas , magnetic fields may organize as coherent loops at granular and subgranular scales ( @xmath21000  km ; * ? ? ? * ) , that these small loops are dynamic @xcite , that they pervade the quiet solar surface and may even connect with upper atmospheric layers @xcite . yet , this picture is still incomplete . for example , we lack a complete mapping of the full magnetic field vector on extended fields of view , because the linear polarization signals are intrinsically weak ( they are second order on the transverse magnetic field component ) , and high spatial resolution maps on linear polarization are rather patchy @xcite , which has led to incomplete ( and sometimes , physically problematic ) characterizations of the topology of the field @xcite .    here we look for and trace small magnetic loops on extended regions of the quiet sun observed with the highest spatial resolution . loop - like structures are a natural configuration of the magnetic field due to its solenoidal character . while they can be traced as single , individual , coherent entities , they characterize the magnetic field at large , and their statistics and evolution may shed some light on the origin of the very quiet sun magnetism , in particular , on the operation or not of local dynamo action @xcite . on the other hand , the organization of the field at small scales affect the organization of the magnetic field at larger scales and in higher atmospherics layers @xcite , and dynamics @xcite . we find evidence for the small scale loops appearing rather irregularly , as in bursts and clumps . moreover , wide regions of the very quiet sun show very low magnetic activity and no apparent sign of organized loops at the detection level of the instruments . these extremely quiet ( _ dead calm _ ) regions are an intrinsic characteristic of the statistical distribution of these events . this paper is devoted to the analysis of disk center quiet sun observations obtained with the imax instrument @xcite onboard the sunrise balloon borne observatory @xcite . imax is a fabry - prot interferometer with polarimetric capabilities at the fei line at 5250.2  . we analyze two different data sets . both have the same properties except that they trace different regions of the quiet sun and were observed at different times , being times series of 22 and 31 min duration . they consist of five filtergrams taken at @xmath640 , @xmath680 and + 227 m from the fei 5250.2  line center . the field of view is 46.8@xmath7 ( 35.6@xmath835.6 mm@xmath1 ) , 20 times larger than the one observed by @xcite . the spatial resolution is of about 0@xmath9.15 - 0@xmath9.18@xmath9 . the time cadence is 32 s ( note that it is 28 s in * ? ? ? * ) , allowing a noise level of @xmath10 and @xmath11 i@xmath12 in the circular and linear polarization , respectively ( i@xmath12 being the continuum intensity ) . we look for small magnetic loops in the data set : coherent structures that appear as a dipole in the longitudinal magnetogram ( i.e. , adjacent positive and negative patches in stokes-@xmath13 maps ) , and a linear polarization patch between them ( see figure 1 ) , and that remain identifiable for several ( at least 2 ) time frames ( @xmath141 minute ) . we looked for small magnetic loops by direct visual inspection . a systematic search was performed in both field - of - views ( fovs ) at all times , for these structures by one of the authors ( mjmg ) , recording the position and evolution at different times of every single event . the method was validated by selecting a small area ( @xmath15  mm@xmath1 ) and independently looking for such structures by a different observer ( rms ) . in this control region , the first observer found @xmath16 loops , the second one @xmath17 . @xmath18 of them were found by both observers . from those values , the actual number of loops @xmath19 on the area may be estimated by the laplace ratio ( or lincoln - petersen index ) for population estimates : @xmath20 ( e.g. , * ? ? ? * ) ; a better , less biased , estimate being @xmath21 @xcite . the variance of this estimate is @xmath22=4.9 $ ] @xcite . these formulae have been used in the literature to estimate the size of ecological populations ( e.g. , * ? ? ? * ; * ? ? ? * ) , and the number of errors in a message @xcite.the main difficulty for applying the method to our case is that the objects to be counted ( the loops as defined in the first paragraph of this section ) may not be unambiguous . to guarantee that this condition is fulfilled ( i.e. , that both observers were identifying and counting the same objects ) , both observers went again over all the structures found in the area by both , and they had to agree over all the events to be counted as loops . from this analysis we conclude that the total number of magnetic structures that we find in the whole dataset could be understimated by @xmath335% . we note that the most clear cases ( those with the strongest polarization signals , lasting longer , and clearly isolated from neighbouring magnetic patches ) were often found by both observers in the control area . the 35% discrepancy is due to those events found by one observer but not the other ; these cases correspond to the most subtle ( often bordering the detection limit ) events . we identify 497 small magnetic loops emerging in the observed regions . taking into account the total time of observation and the spatial area covered , this amounts to an emergence rate of small dipoles of 0.25  loops  h@xmath4  arcsec@xmath5 , rising up to 0.34  loops  h@xmath4  arcsec@xmath5 when we apply the correction for undetected ( although present in the data ) loops given above . these values are one order of magnitude larger than the previous estimation found by @xcite , but compatible with mart ' inez gonzlez ( 2007 ) and mart ' inez gonzlez et al . there are several reasons for the discrepancy . first , the former study covered a relatively small area and , if the emergence is not strictly uniform ( as we shall discuss below ) , the rate can be greatly understimated . this fact was already pointed out in @xcite , where it was noted that there was evidence for preferential emergence regions . the reason why the studies in the near - ir and this letter are in agreement is because , either having a better zeeman sensitivity ( mart ' inez gonzlez et al . ( 2007 ) and mart ' inez gonzlez et al . ( 2010 ) used the more sensitive most importantly to linear polarization spectral line of fe  i at 1.5  @xmath23 m ) or having a better spatial resolution makes both studies more sensitive , inducing the identification of more ( weaker and/or smaller ) structures in the field . both opposite polarity feet and the linearly polarized bridge between them were tracked during the full loop phase . in most cases ( 60% of the events ) , the loop collides or merges to some degree with a neighbouring structure and we could not trace further its individual evolution . this is a common case because at our level of sensitivity and at such spatial resolution , most of the pixels in the fov show circular polarization , and many , both linear and circular . for the remaining 40% of the cases , the loop individual history could be traced beyond ( and before ) their complete loop phase ( i.e. , when both three polarization patches are seen simultaneously ) . in nearly half of these cases ( 56% ) , both footpoints and linear polarization appear and/or disappear simultaneously because the structure falls below the detection level of the instrument ( or it submerges ) . we named this population of loops as `` low - lying '' ( see * ? ? ? * ) . this population of loops is very particular of the quiet sun . in 42% of the cases , linear polarization precedes the detection of both footpoints and then disappears before them too . @xcite computed the line - of - sight velocity of the loops using the stokes @xmath13 zero - crossing shift and verified that all the loops having this very same time evolution were rising @xmath24-loops . we have no reasons for thinking that the loops found in imax are a different population . however , we can not compute the stokes @xmath13 velocity in the imax data sets analyzed in this paper , hence , the identification of these loops as rising @xmath24-loops can be put in doubt . in just 3 instances ( 2% ) , we found that first the opposite polarities appear , then the linear polarization between them , and everthing disappears , wich could be interpreted as the emergence of a u - loop , or , more probably ( considering the local evolution of the flow ) , a submergence of an @xmath24-loop . figure [ ejemplos ] display four examples of small loops found in the imax data . note that , for the sake of clarity , we have only drawn the contours of interest , avoiding circular and linear polarization patches adjacent to the loop structure . the time runs from top to bottom , the time cadence being irregular . the first and third columns represent typical loops in which the linear polarization disappears at some point in the evolution of the loop while the footpoints stay in the photosphere ( probably rising @xmath24-loops ) . these two loops have , however , some peculiarities that differentiate them . the loop in the first column appears at the border of an expanding granule . as the granule expands , the entire loop is dragged by the plasma flow . the loop in the third column contains a linear polarization signal with a gap in between . these two peculiarities are a consequence of the spatial resolution of the imax data that allows us to trace the dynamics of the linear polarization . the loop in the second column of fig . [ ejemplos ] has another linear polarization patch with a curious dynamics . it seems that , somehow , the negative footpoint is disconected from the positive one ( i.e. , the loop breaks ) and that it connects somewhere else to the right of the positive footpoint . of course , it is just a visual impression . the example in the fourth column is an example of two loops appearing very close in time and in space . they also disappear more or less at the same time , hence , one could think this is an evidence of a sea - serpent magnetic field line . note also that the rightmost loop rotates . all the observed dipoles are smaller than @xmath25  mm ( center - to - center distance between the two opposite polarity patches ) , becoming increasingly more abundant at smaller scales , with most of the observed dipolar structures being @xmath26  mm ( figure 2a ) . this is barely three times the spatial resolution limit of imax , which suggests that the detection and our statistics might be limited by the instrument . the tilt angle of these dipoles is uniformly distributed , meaning that it does not follow the hale s polarity law . this last result is consistent with @xcite and even with the behaviour of the smallest ephemeral regions . figure 2b displays the distance between footpoints with respect to the separation between them at the initial time . therefore , positive values mean that the footpoints separate with time and negative ones indicate that the footpoints approach each other . in average , the distance grows linearly with time with a velocity of @xmath27 km s@xmath4 , comparable to typical granular values . this indicates that the loops passively follow the granular flows , as expected from weak magnetic features @xcite . the magnetic properties of these small dipoles are represented in fig . 2c and 2d . they are obtaining inverting the data in the weak field approximation following @xcite . the magnetic flux has been computed in the area containing the observed signal . the frontier has been defined by eye ( as an isocontour of magnetic flux density ) and hence is slightly different for the different structures . the magnetic flux density is the mean value of the magnetic flux densities in this same region . the magnetic flux of the loops can be explained with a power law using the exponent found by @xcite . this means that the population of small dipoles follows the population of magnetic fields in the quiet sun . but looking at the histogram of magnetic flux densities , the loops are located at the end tail of the histogram ; it is mostly in the range 10 - 20  mx  cm@xmath4 i.e . , 10 - 20  g if the magnetic field were uniformly distributed and volume filling . this is compatible with the results of @xcite who state that , in the quiet sun , the larger the signal the larger the degree of organization of magnetic fields . although small scale loops are found all over the observed areas , their spatial distribution does not seem to be completely uniform ( fig . it can be observed that , at some locations , loops appear repeatedly and succesively as in bursts , forming clusters , a behaviour that has been noticed before by mart ' inez gonzlez & bellot rubio ( 2009 ) , who pointed out that , often , the appearence of loops made it more likely that new ones were later detected nearby . on the other hand , extended areas seem to be noticeably empty of such events , as if voids appeared in the distribution . however , this may be deceiving since voids are also formed even in strictly uniform distributions of points @xcite . a quantitative analysis to determine if these voids are statistically significant was performed . this requires on the first place , an unambiguous definition of ` void ' , a non - trivial task in itself ( see e.g. , * ? ? ? * ; * ? ? ? * and discussions therein ) . we adopted the simplest definition here and considered only voids of circular shape : the largest empty circle that can be fitted in a given region of the point field equivalently , an empty circle limited by three points of the distribution . if the loops appeared uniformly on the solar surface , then the probability of finding a void of area between @xmath29 and @xmath30 within the field of view ( square surface of area @xmath31 ) would be ( see appendix ) : @xmath32 where @xmath33 , @xmath34 is the surface density of points , and @xmath35 is the total number of voids espected in the fov , which is given by equation  ( a1 ) . for the two data sets studied here @xmath36  mm ( which is slightly smaller than the nominal fov because we have excluded the apodized exterior area ) , and @xmath37  mm@xmath5 . note that we only need a single value of @xmath34 since the number of loops detected in both data sets are very similar ( i.e. , 248 and 249 events ) . figure 4 shows the number of voids per unit area in both datasets and for the corresponding poisson distribution . except for the smallest areas , the distribution of large circular voids in the first dataset is not significantly different from the uniform one . on the second data set , however , apart from the overabundance of small scale voids , large circular voids seem to be significatively more abundant than a strictly uniform distribution would suggest . actually , for the parameters found for these observations , the probability of finding a circle with an area lager than @xmath38  mm@xmath1 ( equivalently , diameter larger than @xmath39  mm ) is very low ( @xmath40 ; see the inset plot in fig . we conclude , therefore , that there is statistical evidence for the two voids marked in figure  3a to be real and not due to chance . moreover , the oberabundance of small voids is interpreted in terms of a clumpy structuring ( see how loops appear in clumps , like a gurgle phenomena , in fig 3 ) .    in order to relate the distribution of loops ( and the voids ) to the global magnetism in the observed area , fig . 3 shows the integrated longitudinal ( in black and white ) and transverse ( in yellow ) magnetograms for the two observations . when dealing with the linear polarization , one has to remind that it is a biased estimator of the transversal field component . in the plot , this bias is statistically partially removed as follows : we compute the bias for a percentile 95 when the observations are pure noise ( * ? ? ? * see ) ( note that this bias value depend on each pixel , i.e. , on the actual intensity profile ) . this means that the `` real '' transverse magnetic will be below this bias value with a probability of 95% . we have decided to put all the values smaller than this bias to 0 . figure 3 shows only the statistically significant patches of linear polarization appearing at all the observed times . the positions of the loops do not seem to be clearly correlated with the longitudinal magnetogram , but the voids encircled by the loops show less magnetic activity than other areas in the fov . considering the correlation with linear polarization , it suggests that most of the linear polarization signals that are detected are associated to loop structures embeded in the formation region . it is known that even in very quiet areas of the sun magnetic fields may organize naturally forming loops at granular scales . in this study we extended this observation to the smallest spatial scales observable ( 100 - 1000  km ) , finding an increasing number of loops at smaller scales up to the resolution limit . this finding suggests that the organization of magnetic fields might continue beyond that limit . we can not reconstruct the complete magnetic field topology because 1 ) the finite spatial resolution of our observations is ( perhaps inherently ) above the organization scale of the magnetic fields , and 2 ) we lack linear polarimetric sensitivity , which gives us only fragmentary information on the transversal ( horizontal ) component of the magnetic fields . due to these limitations the loop structures that we observe are biased towards relatively large and relatively strong with respect to the magnetic flux density in the neighbouring areas . we found evidence that the loops thus detected are not randomly distributed on the solar surface , but rather that they may appear in bursts , and that they are noticiable absent from extended areas which are , also , only weakly magnetized . it is not yet clear what is the nature of the magnetic fields in the quiet sun what are the fundamental physical mechanisms involved in their generation and evolution . the presence of these _ dead calm _ areas in the quiet sun ( and small scale loops hotspots ) represent an important constrain on the origin of magnetic fields in the very quiet sun and on the dominant dynamic and magnetic mechanisms taking place there . it is thought that the magnetism of the quiet sun can be the result of the emergence of underlying organized magnetic fields @xcite or the dragging of the overlying canopy fields @xcite . it would then be necessary to understand why there are emergence hotspots and dead calm areas . another possibility is that they are just recycling of the decay of active regions as they diffuse and migrate to the poles . but it seems unlikely that such random walking would lead to the kind of organized structures and to the spatial patterns reported here . it is also possible that they are linked to some type of dynamo action taking place in the solar surface @xcite . it seems now clear that coherent velocity patterns are a requisite for dynamo action to take place @xcite . the most obvious coherent velocity pattern in the solar surface is granulation . if this velocity pattern is involved in some dynamo action , it is reasonable that it forms coherent magnetic structures ( such as the loops ) , although these does not need to be organized at the same granular scales ; it could well be that they form intermittent patterns as the ones observed here . actually , theoretical considerations @xcite and laboratory experiments @xcite support the idea that the onset of turbulent dynamo action may be highly intermittent and bursty . finally , it could just be that these small scale loops represent the far tail of a continous range of structures from a global dynamo , just lying at the other end from sunspots . their spatial statistics would then reflect the velocity patterns on the last ( shallowest ) layers of magnetic field emergence . future models that we construct to understand the generation of magnetic fields in the very quiet sun have to explain the spatio - temporal coherences that we report . further work is needed to extend these results in larger areas of the sun and along the solar cycle . we specially thank dr . juan betancort rijo for very helpful discussions on the determination of the statistics of voids , which have improved the paper and strengthen the conclusions . we also thank valent ' in mart ' inez pillet for helpful discussions . eliot hijano enjoyed a summer research grant at the instituto de astrof ' isica de canarias . financial support by the spanish ministry of economy and competitiveness through projects aya2010 - 18029 ( solar magnetism and astrophysical spectropolarimetry ) and consolider - ingenio 2010 csd2009 - 00038 is gratefully acknowledged . we derive equation ( 1 ) adapting to our case some arguments of the strategy by @xcite . the probability that three points @xmath41 distributed at random ( uniformly ) on an square with area @xmath42 , have coordinates between @xmath43 and @xmath44 , @xmath45 and @xmath46 ( @xmath47 and @xmath48 ) , is @xmath49 . with the change of variables @xmath50 , @xmath51 , we may express the probability for the center of circumcircle of the three points to lie between @xmath52 and @xmath53 , @xmath54 and @xmath55 , its radius between @xmath56 and @xmath57 , and the azimutal angles of each point between @xmath58 and @xmath59 , then , as @xmath60 , where @xmath61 is the determinant of the jacobian matrix . the probability of one such a circle to lie between the bounds of the large square and its radius to be between @xmath56 and @xmath57 is @xmath62 , where @xmath52 and @xmath54 have been integrated betwen @xmath56 and @xmath63 , and @xmath58 between 0 and 2@xmath64 , taking into account that the integral of the angular part of @xmath65 is @xmath66 . alternatively , the probability density of the area @xmath33 of such a circle is @xmath67 .    on the other hand , in a homogeneous poisson field with density @xmath68 ( @xmath69 points randomly distributed over an area @xmath70 ) , the probability of finding @xmath71 points in a region of area @xmath29 is @xmath72 . therefore , the probability of the circumcircle of three points to be void is @xmath73 . finally , @xmath69 points determine @xmath74 ( @xmath75 ) different triplets ( hence , possible circles ) , and the number of void circles with areas between @xmath29 and @xmath30 within the square bounds is @xmath76 . the total number of void circles is : @xmath77,\ ] ] where @xmath78 and @xmath79 is the error function @xcite . the probability of voids of area @xmath29 with our constrains is then the number of voids of area @xmath29 divided by the total number : @xmath80      we specially thank dr . juan betancort rijo for very helpful discussions on the determination o f the statistics of voids , which have improved the paper and strengthen the conclusions . the c ontribution of eliot hijano to this work was in the frame of the summer grants offered by the instituto de astrof ' isica de canarias . financial support by the spanish ministry of economy and competitiveness through projects aya2010 - 18029 ( solar magnetism and astrophysical spectrop olarimetry ) and consolider - ingenio 2010 csd2009 - 00038 is gratefully acknowledged .                          , j.  m. , pearce , f. , foster , c. , platen , e. , brunino , r. , neyrinck , m. , basilakos , s. , fairall , a. , feldman , h. , gottlber , s. , hahn , o. , hoyle , f. , mller , v. , nelson , l. , plionis , m. , porciani , c. , shandarin , s. , vogeley , m.  s. , & van de weygaert , r. 2008 , , 387 , 933        , s. , beeck , b. , pietarila , a. , schssler , m. , solanki , s.  k. , martnez pillet , v. , bonet , j.  a. , del toro iniesta , j.  c. , domingo , v. , barthol , p. , berkefeld , t. , gandorfer , a. , knlker , m. , schmidt , w. , & title , a.  m. 2010 , , 723 , l149          , r. , & tsuneta , s. 2009 , in astronomical society of the pacific conference series , vol . 415 , the second hinode science meeting : beyond discovery - toward understanding , ed . b.  lites , m.  cheung , t.  magara , j.  mariska , & k.  reeves , 132                          , v. , del toro iniesta , j.  c. , lvarez - herrero , a. , domingo , v. , bonet , j.  a. , gonzlez fernndez , l. , lpez jimnez , a. , pastor , c. , gasent blesa , j.  l. , mellado , p. , piqueras , j. , aparicio , b. , balaguer , m. , ballesteros , e. , belenguer , t. , bellot rubio , l.  r. , berkefeld , t. , collados , m. , deutsch , w. , feller , a. , girela , f. , grauf , b. , heredero , r.  l. , herranz , m. , jernimo , j.  m. , laguna , h. , meller , r. , menndez , m. , morales , r. , orozco surez , d. , ramos , g. , reina , m. , ramos , j.  l. , rodrguez , p. , snchez , a. , uribe - patarroyo , n. , barthol , p. , gandorfer , a. , knoelker , m. , schmidt , w. , solanki , s.  k. , & vargas domnguez , s. 2011 , , 268 , 57            , f. , berhanu , m. , monchaux , r. , aumatre , s. , chiffaudel , a. , daviaud , f. , dubrulle , b. , bourgoin , m. , odier , p. , plihon , n. , pinton , j .- f . , volk , r. , fauve , s. , mordant , n. , & ptrlis , f. 2008 , physical review letters , 101 , 074502
we analyze two regions of the quiet sun ( @xmath0  mm@xmath1 ) observed at high spatial resolution ( @xmath2100  km ) in polarized light by the imax spectropolarimeter onboard the sunrise balloon . 
 we identify 497 small - scale ( @xmath3400  km ) magnetic loops , appearing at an effective rate of 0.25  loop  h@xmath4  arcsec@xmath5 ; further , we argue that this number and rate are underestimated by @xmath330% . however , we find that these small dipoles do not appear uniformly on the solar surface : their spatial distribution is rather filamentary and clumpy , creating _ 
 dead calm _ areas , characterized by a very low magnetic signal and a lack of organized loop - like structures at the detection level of our instruments , that can not be explained as just statistical fluctuations of a poisson spatial process . 
 we argue that this is an intrinsic characteristic of the mechanism that generates the magnetic fields in the very quiet sun . 
 the spatio - temporal coherences and the clumpy structure of the phenomenon suggest a recurrent , intermittent mechanism for the generation of magnetic fields in the quietest areas of the sun .