Long-Term Evolution of Three Light Bridges Developed on the Same Sunspot
Ana Belén Griñón-Marín, Adur Pastor Yabar, Rebecca Centeno, Héctor Socas-Navarro
©© L ong -T erm E volution of T hree L ight B ridges D eveloped on the S ame S unspot A. B. Griñón-Marín , , , A. Pastor Yabar , R. Centeno , and H. Socas-Navarro , Instituto de Astrofísica de Canarias, Vía Láctea, 38205 La Laguna, Tenerife, Spain Universidad de La Laguna, Departamento de Astrofísica, 38206 La Laguna, Tenerife, Spain W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA Institute for Solar Physics, Department of Astronomy, Stockholm University, Albanova University Centre, SE-106 91 Stockholm,Sweden High Altitude Observatory (NCAR), 3080 Center Green Dr. Boulder CO 80301February 2021
ABSTRACT
One important feature of sunspots is the presence of light bridges. These structures are elongated and bright (as compared to the umbra)features that seem to be related to the formation and evolution of sunspots. In this work, we studied the long-term evolution and thestratification of di ff erent atmospheric parameters of three light bridges formed in the same host sunspot by di ff erent mechanisms. Toaccomplish this, we used data taken with the GREGOR Infrared Spectrograph installed at the GREGOR telescope. These data wereinverted to infer the physical parameters of the atmosphere where the observed spectral profiles were formed of the three light bridges.We find that, in general, the behaviour of the three light bridges is typical of this kind of structure with the magnetic field strength,inclination, and temperature values between the values at the umbra and the penumbra. We also find that they are of a significantlynon-magnetic character (particularly at the axis of the light bridges) as it is deduced from the filling factor. In addition, within thecommon behaviour of the physical properties of light bridges, we observe that each one exhibits a particular behaviour. Anotherinteresting result is that the light bridge cools down, the magnetic field decreases, and the magnetic field lines get more inclinedhigher in the atmosphere. Finally, we studied the magnetic and non-magnetic line-of-sight velocities of the light bridges. The formershows that the magnetic component is at rest and, interestingly, its variation with optical depth shows a bi-modal behaviour. For theline-of-sight velocity of the non-magnetic component, we see that the core of the light bridge is at rest or with shallow upflows andclear downflows sinking through the edges.
1. Introduction
In recent years, our understanding of the formation and evolutionof sunspots has improved thanks to, among other things, the ad-vent of spectropolarimetric observations and / or high temporal,spatial, and / or spectral resolution. Light bridges (LBs; Brayley1869) play an important role in our understanding of the evolu-tionary stages of sunspots. LBs are irregular, bright, elongatedstructures that are seen in umbra during the formation and / ordecay of sunspots or pores (Sobotka 2003, Thomas & Weiss2004). They can indicate the break-up of sunspots in the decay orthe formation phases of complex active regions, as if they were‘seams’ where a sunspot forms or decays (Garcia de La Rosa1987). Their lifetimes are shorter than those of the sunspots thathost them, and they are very dynamic. LBs have di ff erent shapesand sizes, varying from less than 0 (cid:48)(cid:48) . Article number, page 1 of 27 a r X i v : . [ a s t r o - ph . S R ] F e b & A proofs: manuscript no. aanda
Currently, there is no established consensus on how LBs areformed, but some authors have studied possible causes. Vázquez(1973) proposed a first interpretation of photospheric LBs andreported that they were produced by sunspot decay precedingthe restoration of the granular surface. In another scenario, Kat-sukawa et al. (2007) observed a mature sunspot and found thatmany umbral dots appear in the umbra of the sunspot to form anLB that cuts across the umbra. Toriumi et al. (2015a) proposedthat LBs of convective origin are formed due to the emergenceof the horizontal magnetic field inside an umbra.One physical property that makes these structures di ff erentfrom the umbra is the magnetic field, which is weaker and morehorizontal in LBs than in their surrounding umbra (Beckers &Schröter 1969; Lites et al. 1991; Leka 1997; Jurˇcák et al. 2006;Louis et al. 2015). Leka (1997) proposed the presence of a mag-netic canopy structure above LBs by carrying out the first sys-tematic study analysing 11 LBs with full spectropolarimetricdata. Later, Jurˇcák et al. (2006) confirmed this model by givingthe stratification of plasma parameters with optical depth. Theysuggested that the field-free plasma intrudes vertically into theumbra from below the photosphere and forms a magnetic fieldcanopy configuration (see Figure 7 from Jurˇcák et al. 2006). Thiswas the first study to analyse the atmospheric parameter stratifi-cations of two LBs. They found that the magnetic field strengthis weaker in the deep layers of LBs, and that the inclination ofthe magnetic field lines decreases with optical depth, acquiringsimilar values of the surrounding umbra at greater optical depths.Finally, Toriumi et al. (2015a) supported their interpretation us-ing the numerical results of the radiative magnetohydrodynamicssimulation of a large-scale flux emergence.Regarding the LB velocity structure, many studies have re-ported blueshifts (Beckers & Schröter 1969; Hirzberger et al.2002; Schleicher et al. 2003; Katsukawa et al. 2007; Rimmele2008) with respect to the umbral velocities, others have mea-sured redshifts (Rüedi et al. 1995; Shimizu 2011), and yet othersa combination of both blueshifts and redshifts (Leka 1997; Rim-mele 1997; Bharti et al. 2007; Giordano et al. 2008; Rouppe vander Voort et al. 2010; Toriumi et al. 2015b; Guglielmino et al.2017; all these papers reported upflows in the middle part of theLB and downflows at the edge). Louis et al. (2009) reported, forthe first time, supersonic downflows with values of up to 10 km / sin the photosphere. They analysed complex Stokes V spectrawith double red lobes with two-component inversions. Further-more, Lagg et al. (2014) measured supersonic downflows at theedges of an LB granule with subsonic upflows in the middle ofit. Rimmele (1997) found a correlation between the brightnessand upflow velocities in strong granular LBs, considering thisevidence of the magnetoconvective origin of this kind of LB.One feature discovered in recent decades related to LBs isthe presence of dark lanes running parallel to their axis. Liteset al. (2004) found that the optical depth of this lane is in therange of 200–450 km above the horizontal plane defined by theumbral floor. Moreover, Berger & Berdyugina (2003) measuredthe width of a dark lane and obtained a value of 380 km. Falcoet al. (2016) calculated the main axis of the LB to have an aver-age width of ≈
225 km and a length of about 6000 km. These darklanes can be explained by the same principle that produces darkcores in bright penumbral filaments (Heinemann et al. 2007;Ruiz Cobo & Bellot Rubio 2008), and they are a common fea-ture of strong LBs (Berger & Berdyugina 2003). Giordano et al.(2008) and Rouppe van der Voort et al. (2010) measured upflowvelocities surrounded by downflows along the central dark lane.In the case of Plateau LBs, the dark lane has a small, transverse,Y-shaped dark lane similar to dark-cored penumbral filaments (Schlichenmaier et al. 2016). This feature is not seen in granularLBs.In May 2014, four main LBs were formed in the leadingsunspot of the NOAA active region 12049. We took advantageof one of the new 1.5-m class solar telescopes to address, forthe first time, a temporal evolution (the timescale of the life ofa sunspot, as some previous studies had already performed sucha temporal study of an LB but on scale of hours, i.e. Shimizu2011) of some atmospheric parameters, both in space and theoptical depth stratification of three of them, using spectropolari-metric data.
2. Observations and methodology
We present an analysis of three LBs observed in the leadingsunspot of NOAA AR12049. This sunspot was observed from2014 May 2 to 2014 May 5. The data were taken using theGREGOR Infrared Spectrograph (GRIS; Collados et al. 2012)installed on the GREGOR (Volkmer et al. 2010; Schmidt et al.2012; Soltau et al. 2012) solar telescope at Observatorio delTeide (Spain). The observed spectral range was centred on theFe i (cid:48)(cid:48) . ff erent orientation. Additionally, the image(slightly) rotates from the beginning to the end of the scan. Sincethe time needed for the acquisition of the maps were short andno data were taken when the image rotated the most, the rotationfrom the beginning until the end of the scan was kept below afew degrees (see last column of Tab. 1). Figure 1 shows the con-tinuum intensity maps of all the scans analysed. In this figure, allthe scans are orientated such that north is at the top of the page,using the apparent X and Y solar coordinates calculated by Dr.M. Franz, who aligned them using Helioseismic and MagneticImager (HMI; Scherrer et al. 2012, Schou et al. 2012) data.The time coverage of the LBs with GRIS data is rather poor.A better time coverage of the spot was achieved by the HMIinstrument (which continuously records the evolution of the vis-ible surface) on board the Solar Dynamics Observatory (SDO;Pesnell et al. 2012) space mission. In Figure 2, we can see theevolution of NOAA AR12049 and its LBs as it crossed the vis-ible solar disc. From this sequence, it seems that each LB wasformed as a consequence of di ff erent processes, and develops ina di ff erent way.Light Bridge 1 was formed due to a strong penumbral pro-trusion into the umbra. This protrusion starts to intrude into theumbra on May 2. This structure gradually grows, splitting theumbra into two parts. This process takes around two days, untilMay 4. This LB is the largest and widest of all the LBs hosted bythis sunspot, and it is present until the end of the available timesequence, when the spot crosses the western limb of the Sun (seeFigure 2).In contrast, the other LBs that appear in the sunspot are quitethin and their lifetimes are shorter. LB2 was a weak intrusion ofthe penumbra into the umbra that took place to the west of LB1.This intrusion began on May 2 and reached the opposite side of http: // archive.leibniz-kis.de / pub / gris / yyymmdd / context_data / whereyyyy is year, mm is the month, and dd is the day of the observation.Article number, page 2 of 27ame(s) of author(s): short title Table 1.
Details of the observations taken with the GRIS spectrograph.
Date Hour (UT) Scan arcsec ) Max. Rot. Ang. (deg)2014 May 2 11:35 2_01 12.365 10 27.00 x 63.45 5.342014 May 2 15:05 2_02 9.701 10 40.50 x 63.32 2.832014 May 3 09:59 3_03 4.620 5 33.75 x 63.45 0.542014 May 3 10:49 3_04 5.431 5 50.09 x 63.45 3.132014 May 3 14:05 3_05 5.957 5 54.00 x 63.72 7.182014 May 3 17:41 3_06 8.085 5 43.20 x 63.32 0.432014 May 5 09:07 5_07 29.05 10 40.50 x 63.59 0.11 Notes.
The rows represent the di ff erent observed maps. The first two columns correspond to the date and time of the observations, the third columnis the number of the scan (hereinafter, this is the notation used to refer to each scan), the fourth is the heliocentric angle, the fifth and sixth columnscorrespond to the number of accumulations and the number of the steps used for each observation, respectively, and the last column correspondsto the rotation angle between the first and last slit positions during the scan. Fig. 1.
Data observed by GRIS spectrograph. The details of these ob-servations are summarised in Tab. 1. The labels indicate the observationday and the number of the scan of the third column of Tab. 1 (observa-tionDay_scan). the umbra at the end of the same day; it then started to retreat un-til its disappearance on May 4. In contrast to LB1, this one par-tially dissolved shortly after splitting both umbral cores. In thisprocess, the northern part dilutes and forms an area of ‘bright’umbral core, while the southern part gets wider and brighter. OnMay 4, this remnant of LB2 seems to merge with the southernpart of LB1.Light Bridge 3 formed as two umbrae approached each other.On April 29, the leading sunspot of NOAA active region 12049had a roundish shape with a full umbra (with no LBs). Somesmall pores were starting to approach it from the northeast. Asthe pores merged with the sunspot, it lost part of its penumbra,and a bright area appeared between the spot and pores (fromApril 30 until May 1). As this bright area became narrower, iteventually led to the birth of LB3, which continued to get nar-rower as the two umbrae got closer and finally merged on May 4(starting from the northern part of the LB and later extending tothe south). The sunspot developed a fourth LB, but since it wasnot observed with GRIS, we did not analyse it in this study. Ta- ble 2 summarises the properties of the three LBs that we analysein the coming sections.
Table 2.
Summary of the formation and evolution of the LBs.
Light Bridge Formation & EvolutionLB1 Strong and broad penumbral intrusionthat breaks the umbra into two partsLB2 Weak and thin penumbral intrusion,which, after splitting the umbra, retreatsuntil its disappearanceLB3 Close approach of two umbra until theLB disappears from the northern part andfinally from the south
After recording the Stokes parameters with GRIS, the softwaresupplied by the instrument developers was applied (Collados etal., in prep.). With this dedicated software, the dark current wassubtracted, the flat-field and the bad pixels were corrected, thedata were demodulated, and the instrumental cross-talk was cor-rected. However, after applying these routines, certain featuresneeded to be considered before proceeding with the data analy-sis. The first step was to correct the continuum intensity trendsof the data. To do so, we used the preceding and the followingflat-field images. We reduced them as though they were scienceproducts, and we obtained the intensity-averaged spectra of eachflat-field map. These two averaged profiles were then fitted witha polynomial of order 13, avoiding spectral lines as these mayworsen the fit of the continuum. We interpolated the intensity tobe corrected from the times at which the flat fields were takento the time when each slit position of the data was recorded.Finally, we corrected the data (the four Stokes parameters), di-viding each observed Stokes profile (including Q , U , and V ) bythis interpolated intensity correction.In the next step, we performed the wavelength calibration ofthe data by fitting a second-order polynomial to the core of thesame lines in both the Fourier transform spectrometer atlas (Liv-ingston & Wallace 1991) and the flat-field averaged-intensityspectra. The linear fit between these two sets results in the wave-length calibration of the observed spectral grid.The third step was to remove some high-frequency fringesthat persist even after the flat-fielding correction, mostly in the Article number, page 3 of 27 & A proofs: manuscript no. aanda
Fig. 2.
Temporal evolution of the intensity maps of the leading sunspot of NOAA active region 12049 observed with the HMI instrument. Thereare four main LBs labelled LB1, LB2, LB3, and LB4, which evolve in di ff erent ways. The first three LBs are analysed in this work.Article number, page 4 of 27ame(s) of author(s): short title intensity data. To do that, we applied Fourier passband filteringin the 384–417 mÅ frequency range.While observing, there were sudden and short episodes ofvery bad seeing. For these slit positions, the accuracy of the de-modulation scheme was worse. This is evident in the form ofstrong polarimetric interference fringes in those slit scanningsteps, which are more evident in Stokes Q , U , and V . Since theseslit scanning steps strongly a ff ect their neighbouring ones duringthe deconvolution process (explained in following section), wesubstituted these columns with an interpolated column obtainedfrom the previous and following scan scanning steps, but theyare not analysed in what follows. When data are recorded, the optical devices in the optical pathmodify the image coming from the Sun. This is usually knownas the telescope point spread function (PSF). In addition, forground-based observations, the Earth’s atmosphere blurs the im-age coming from the Sun. This is usually known as the see-ing PSF and mostly determines the spatial resolution of the ob-servations (even when using adaptative optics). In order to re-move these e ff ects, we deconvolved each map by applying aregularised principal component analysis and using a static PSFwhose performance over space-borne data and methodology isdescribed in Quintero Noda et al. (2015).To calculate the PSF of the optical system (for both the tele-scope and the Earth’s atmosphere), we followed Collados’s pro-cedure (GREGOR internal report) developed with the spectropo-larimetric data taken in the 1.56 µ m range during the transit ofMercury on 2016 May 9. In order to get the PSF, an ideal Mer-cury image was convolved by a 2D Gaussian given by φ ( r ) = α G ( r , σ ) + (1 − α ) G ( r , σ ) , (1)where G and G are two Gaussians with the standard deviationgiven by σ and σ , respectively. G corresponds to the spatialresolution of the telescope (including the seeing), and G ac-counts for the instrumental stray light. α is a constant and definesthe relative weight of each Gaussian. The parameters α , σ , and σ were obtained following a non-linear fit to obtain α = . σ = (cid:48)(cid:48) . σ = (cid:48)(cid:48) . α and σ as obtainedfor the Mercury transit. Regarding to obtain the σ value, weneeded to know the seeing value of our observation day. How-ever, during the observations of the sunspot under study, therewere communication problems between the GREGOR Adapta-tive Optics system and the instrument computer. For this reason,the data header did not record the seeing value ( r ) of our ob-served scans. That is why, in order to use an r close to the realone, we deconvolved each scan using di ff erent seeing values tocalculate the PSF and visually choose the one that did not pro-duce any unusual artefact. Figure 3 shows the images of the samescan deconvolved using di ff erent seeing values to calculate thePSF profile. The number that appears in each map correspondsto the r value (at 1.56 µ m of wavelength) used for the decon-volution. The deconvolved maps with r < .
04 cm seem tosu ff er an over-correction; thus, in order to be conservative, the r = .
77 cm ( r = .
06 cm at 500 nm of wavelength) waseventually selected as the seeing value to be used to calculate thePSF shape. Once the r value of the observations was estimated,and assuming that the first Gaussian ( G ) gave the angular reso-lution, we were able to calculate σ following Equations 2 and3: Fig. 3.
Deconvolved monochromatic images of the same scan for dif-ferent values of seeing Gaussian (given by the lower number in eachpanel). θ ≈ λ obs r (cid:48) ≈ FWHM , (2)where θ is the angular resolution and 206265 is the unit conver-sion to give the result in seconds of arc. Bearing in mind that FWHM = √ ln σ , we can calculate σ as σ = λ obs r (cid:48) √ ln , (3)where λ obs is the central wavelength of this study (1 . µ m ),and r (cid:48) is the seeing value for our spectral range (70 .
77 cm). Thisway, we obtained σ = (cid:48)(cid:48) . Article number, page 5 of 27 & A proofs: manuscript no. aanda
Once α , σ , and σ variables were calculated, the ob-served spectropolarimetric maps were deconvolved using a prin-cipal component analysis regularisation (Ruiz Cobo & AsensioRamos 2013) using a PSF given by Eq. 3. Figure 4 shows thecomparison of the Stokes parameters before and after applyingthe deconvolution. When the maps are ready to use, in order to infer the magneticand thermodynamic properties of the solar atmosphere, wherethe observed spectral lines were formed, we used the Stokesinversion based on response functions (SIR; Ruiz Cobo & delToro Iniesta 1992) code. This code assumes local thermody-namic equilibrium (LTE) to solve the radiative transfer equa-tion for polarised light in order to compute the synthetic Stokesprofiles. SIR computes perturbations in the physical quantitiesat specific locations across the optical depth grid called nodes,and then carries out an interpolation to yield values at all gridpoints. Finally, the code compares the synthetic with the ob-served Stokes profiles, modifying an initial atmospheric modeluntil the di ff erences between the observed and synthetic profilesare minimised.The inversion code returns the optical depth stratifications oftemperature ( T ), magnetic field strength ( | B | ), magnetic field in-clination with respect to the line of sight ( γ los ), magnetic fieldazimuth on the plane of the sky ( φ los ), line-of-sight velocity( v los ), micro-turbulence velocity ( v mic ), electronic pressure ( P e ),gas pressure ( P g ), and density ( ρ ). The last two quantities are de-rived from the temperature and electronic pressure stratificationsafter assuming hydrostatic equilibrium. Moreover, the macro-turbulence velocity ( v mac ), the filling factor ( f f ) of the magneticcomponent, and the stray light fraction ( sl ) might be provided bythe code.In the observed spectral range, there are some spectral linesthat are sensitive to the magnetic field, and here we use two neu-tral iron spectral lines, 15648.515 and 15662.018 Å, with respec-tive Landé factors of 3.0 and 1.5. We avoided the other spectrallines in the observed spectral window either because the atomicparameters were not well characterised (i.e. the 15631.948 Åspectral line) or because of contamination by molecular blendsthat clearly appear in lower temperature regions, mostly in um-bra pixels (i.e. the 15645.016 and 15652.881 Å spectral lines).The atomic parameters of the spectral lines inverted simultane-ously are listed in Tab. 3.We followed di ff erent inversion strategies depending on theinverted area: light bridges (LB), umbra (UM), penumbra (PE),and quiet Sun (QS). ∗ The UM and PE pixels were inverted with one magnetic at-mospheric component by using five nodes for the T , onenode for v mic , three nodes for | B | , and two nodes for γ , φ ,and v los . The atmospheric guess model was the cool umbralmodel by Collados et al. (1994). ∗ For the QS pixels, we followed a slightly di ff erent strategy:all the parameters but T (seven nodes) were constant with op-tical depth and the atmosphere guess model was the Harvard-Smithsonian reference atmosphere model (HSRA; Gingerichet al. 1971). The inclination is measured with respect to the line-of-sight, the 0 ◦ azimuth reference is in the north solar direction, and the azimuth valuesincrease in an anti-clockwise direction. Fig. 4.
Stokes parameter deconvolution. Each pair of panels is amonochromatic map of the Stokes parameter. From top to bottom:Stokes I at the continuum, Stokes Q and Stokes U both at the core of theline, and finally the Stokes V at the red wing. On the left of each pair,the interpolated original map is shown, and on the right is the same mapafter the deconvolution.Article number, page 6 of 27ame(s) of author(s): short title
Table 3.
Characteristics of the spectral lines inverted in this study to obtain the atmospheric parameters of the solar atmosphere where these linesare formed (Borrero et al. 2013, Bloomfield et al. 2007).
Element Ion λ (Å) Exc. Pot. (eV) log(gf) Term α σ ( a o )Fe I 15648.515 5.426 -0.669 D − D F − F Notes.
The first and second columns are the atomic element and the ionisation state of the atom in the inverted spectral lines. The third columncorresponds to the air wavelength of the transition in angstroms, the next column is the excitation potential of the lower level, the fifth column isthe oscillation strength, and the lower and upper levels are listed in the sixth column. The seventh and eighth columns are the Barklem velocityexponential and Barklem cross-section (in units of the Bohr radius squared), respectively. The last two values were calculated following the ABOtheory (Barklem 1998; Anstee & O’Mara 1995) and can be obtained from the tabulated tables in Anstee & O’Mara (1995). ∗ Finally, the LB pixels were inverted with one magnetic at-mosphere and one non-magnetic atmosphere. The nodesused for the inversion of the magnetic component were thesame as for the UM and PE pixels. The atmospheric guessmodel of the magnetic component was again the cool um-bral model. The non-magnetic component was initialised us-ing the atmosphere obtained in the inversion of the averagethe Stokes I profiles coming from the surrounding QS areas.In addition, we only inverted the v los of the non-magneticcomponent (constant with optical depth) and the f f . In otherwords, we used a magnetic component and a non-magneticcomponent with free v los and filling factor. The T was notinverted since the information about this parameter was notenough as we checked from inversion of these pixels withmore complex strategies (allowing the non-magnetic T tovary). This way, the T stratification used was the inferredone in the inversion of the Stokes I of the surrounding QSareas.A list of the free parameters used in the strategies followedfor the di ff erent areas can be found in Tab. 4. Following PastorYabar et al. (2018), the inversion process was repeated 50 timesin each pixel with random initial values of the free parameters( | B | , γ , φ , v los , and v mic ) in a given range (see Tab. 3 of PastorYabar et al. 2018).A final step is required in order to obtain ready-to-use in-ferred parameters. This step involves the resolution of the so-called 180 ◦ ambiguity present in the line-of-sight magnetic fieldazimuth ( φ los ). This ambiguity comes from the fact that a given φ los leads to exactly the same spectral profiles as φ los + ◦ .This ambiguity involves two possible magnetic field topologies,both in the line-of-sight and local reference frame, for each pixel.It is thus mandatory to properly resolve this ambiguity for thewhole observed area in order to study the magnetic field topol-ogy. Here, we handled this step using the technique described byGeorgoulis (2005) and the supplied software. This code returnsthe disambiguated values of azimuth, which together with theinclination of the magnetic field can be transformed to the localsolar reference frame. Hereafter, the inclination ( γ ) is the anglebetween the magnetic field vector and the local solar vertical, theazimuth ( φ ) is on the tangent plane to the solar surface, the 0 ◦ az-imuth reference is in the north solar direction, and the azimuthvalues increases anti-clockwise. We focused our study on T , | B | , γ los , and v los (from magnetic and non-magnetic atmospheres). After inferring the line-of-sight velocity ( v los ), it is necessary tocalibrate it with respect to a reference zero velocity. There aretwo di ff erent ways to do this: absolute and relative calibrations.The absolute calibration can be done using either telluric lines or a laser-based calibration, but we have neither of them. Regardingrelative calibrations, there are two methods that are commonlyused in the bibliography: setting the average QS velocities or themean umbral velocities to zero.We decided to use the relative calibration that sets the umbralaverage velocity to 0. After applying this correction, we realisedthat there was still a residual velocity o ff set with time, thus alongthe scanning direction. This e ff ect was seen most clearly in theQS average velocity along the scanning direction (see black lineof Figure 5). This residual velocity was most probably due toan instrumental problem (private communication with Dr. Col-lados). To solve it, we identified a su ffi ciently large area of QSin the field of view present in all scanning positions. We thenaveraged along the slit the velocities associated with these QSpixels. In this way, assuming that the QS does not vary its av-erage velocity with time, we obtained the velocity variation asthe scan was recorded. Finally, we fitted these variations (onefor each map) with a second-order polynomial (black and redlines of Figure 5, respectively) and used it to correct the velocitytrend. After this trend correction, we applied the umbral rela-tive calibration by calculating the average umbral v los (pink linein Figure 5). These two corrections (the trend produced by theinstrumentation and umbral calibration) were applied to all theoptical depths, leading to ready-to-use velocities. A crucial point is to know the optical depths where the Stokesprofiles are sensitive to the physical parameters of the atmo-sphere, that is, where (and by how much) Stokes profiles changeas one physical parameter is changed at a given optical depth.This is interesting because the inferred stratifications are not re-liable for the whole grid of optical depths, but only for someranges of it. In order to know at which optical depths the ob-served lines are sensitive to perturbations of the physical atmo-spheric parameters, we calculated the response functions (RFs)of each atmospheric parameter analysed in this work for a set ofprofiles representative of PE, UM, and LB. An example of theRFs is represented in Figure 6.One of the analyses in the next section compares the be-haviour of the atmospheric parameters in the UM, PE, and LBs.However, the spectral lines are not sensitive at the same opti-cal depths for these regions. In order to study the parametersin a common set of optical depths for the three areas, we tookfour representative pixels from the UM and from the LB. Wethen calculated the optical depth ranges where each integratedRF contributes 95% of the total area (this contribution is markedwith blue horizontal lines in Figure 6). Finally, we took the op-tical depths that are common to all regions, the LB, the PE, andthe UM. In this case, the observed infrared spectral lines show
Article number, page 7 of 27 & A proofs: manuscript no. aanda
Table 4.
Free parameters used in the strategies followed for the di ff erent regions. Region T | B | γ φ v v T | B | γ φ v v LB 5 3 2 2 2 1 - - - - 1 -UM 5 3 2 2 2 1 - - - - - -PE 5 3 2 2 2 1 - - - - - -QS 7 1 1 1 1 1 - - - - - -
Notes.
The values represent the number of nodes used in the inversion process for each parameter and the superscripts 1 and 2 represent the twocomponents used for the inversions.
Fig. 5.
Velocity calibration: The black line represents the average v los of the QS area of Scan 3_05 along part of the slit used to correct thetrend produced by an instrumentation problem. The red line is a second-degree polynomial that fits the black line. The pink line corresponds tothe value obtained by averaging the velocity of the UM after correctingthe instrumental trend. Finally, the blue line corresponds to the finalresult of the averaged QS v los after applying the previous corrections.The green line marks the 0 value. sensitivity mainly between log( τ ) = − . τ ) = . v los , and the values for the T arebetween log( τ ) = − . τ ) = .
8. Hereinafter, for the rep-resentation of the physical parameters, we use the values shownin Tab. 5.
Maps of the normalised observed continuum intensity with re-spect to the QS, I / I o , temperature, T , magnetic field strength, | B | ,magnetic field inclination, γ (in the local reference frame), andline-of-sight velocity of the magnetic component, v m los (after ve-locity calibration), are shown in Figure 7. All these maps, except I / I o , are the results of the inversions of Stokes profiles averagedin optical depth between the values shown in Tab. 5 (SensitivityRange column) for the magnetic atmosphere , that is, the line-of-sight velocity of the non-magnetic component, v nm los , of the LBareas is not represented in this figure. The temperature stratifica-tion of the non-magnetic atmosphere for each map, together withthe temperature stratification of the HSRA model, are shown inFigure 8 with black and red lines, respectively. The filling factor Fig. 6.
Response functions of the Stokes parameters of the spectral linesused in the inversion of the magnetic field strength (top eight panels) andof the temperature (bottom eight panels). Each map was normalised toits total absolute value. The left line corresponds to the 15648.515 Åspectral line and the right one to 15662.018 Å. The grey scale imagesshow the RFs of the four Stokes parameters, and the plots on the side de-pict the integral of each RF along the wavelength direction. The graphsdepict the integral of each RF in the wavelength direction. The horizon-tal blue lines mark the optical depths for which the area of the integratedRF between these blue lines adds up to 95% of the total, and the greenhorizontal line corresponds to the optical depth where the integrated RFis maximum. maps, f f , and v nm los are analysed in detail and shown in Section3.2.2.At this point, we have our data ready for analysis, which ispresented in the next section.
3. Analysis and results
The main focus of this work is the study of the temporal evolu-tion of the LB atmospheric parameters compared with that of theUM, PE, and QS.
Article number, page 8 of 27ame(s) of author(s): short title
Fig. 7.
Maps of the atmospheric parameters obtained after the Stokes parameter inversions for the magnetic atmosphere averaged in optical depthbetween the values shown in Tab. 5 (Sensitivity Range column). Columns (from left to right): normalised observed continuum intensity withrespect to the QS, temperature, magnetic field strength, inclination in the solar reference frame, and velocity in the line-of-sight reference frame.The rectangles drawn on the intensity maps mark the regions used in the next section to analyse the di ff erent LBs. The white square indicates LB1,the green one is the area taken for LB2, and the black one corresponds to LB3. The contours represent the di ff erent areas taken for analysis of theresults. The black, white, red, blue and pink contours mark the PE, UM, LB1, LB2 and LB3, respectively. The labels indicate the observation dayand the number of the scan of the third column of Tab. 1 (observationDay_scan). Article number, page 9 of 27 & A proofs: manuscript no. aanda
Table 5.
Optical depth ranges used for the analysis of the atmosphericparameters of the LBs.
Sensitivity Range Di ff erencesUpper Layer - Lower Layer | B | (-0.7 , 0.4) (-0.7 , -0.4) - (0.1 , 0.4) γ (-0.7 , 0.4) (-0.7 , -0.4) - (0.1 , 0.4) v los (-0.7 , 0.4) (-0.7 , -0.4) - (0.1 , 0.4) T (-0.5 , 0.8) (-0.5 , -0.2) - (0.5 , 0.8) Notes.
For the figures in which we compare the parameters between dif-ferent layers, we compared the variation of the various physical param-eters considered in two layers at di ff erent averaged optical depths. Theoptical depths used for these averages are shown in the ‘Upper Layer’and ‘Lower Layer’ columns. For the figures in which we study the tem-poral evolution of the atmospheric parameters, we used the values ofthe ‘Sensitivity Range’ column. Fig. 8.
Temperature stratifications. The solid dark line corresponds tothe non-magnetic component for each observed map represented be-tween the optical depth values of the left column of Tab. 5. The reddotted lines correspond to the temperature stratification of the HSRAmodel. | B | , γ , and T In order to better characterise these LBs, we performed threedi ff erent analyses. First, in Section 3.1.1, we study the temporalevolution of | B | , γ , and T (where T = f f ∗ T mag + (1 − f f ) ∗ ( T no − mag ), f f is the filling factor, and T mag and T no − mag are thetemperature of the magnetic and non-magnetic component, re-spectively). To do so, we compared their global behaviour withthat of the UM, PE, and QS. We averaged the physical propertiesin each LB both horizontally and along the line of sight. Thisanalysis distinguishes neither the spatial properties of the LBsnor the optical depth variations, but it allows a preliminary in-sight into the general properties of these LBs. In order to charac-terise the former, we present a second analysis in Section 3.1.2,where we focus on the temporal evolution and spatial distribu-tion of the inferred parameters averaged in optical depth to anal-yse how the LBs change with time and space. Finally, in order tocharacterise the optical depth variation, we compared the di ff er-ence of the LB parameters between the upper and lower layerswith that of the UM and the PE in Section 3.1.3. We computed the spatial and optical depth averages of | B | , γ (in local reference frame), and T for the LB (hereinafter, | B | lb , γ lb , and T lb ), where the magnetic parameters are applicable tothe pixel portion of the corresponding magnetic component, andwhich is detailed later in Section 3.2.1. Furthermore, the tem-perature is a weighted sum of the magnetic and non-magneticcomponents, UM ( | B | um , T um , and γ um ), and PE ( | B | pe , T pe , and γ pe ). We additionally included the QS temperature ( T qs ) in orderto have a standard reference. The pixels taken for each regionare indicated in Figure 7 with black and white contours on thetemperature maps. These calculations are shown in Figure 9.For reference, we first considered the case for the UM, PE,and QS. | B | , γ , and T hardly vary with time during the observedscans and for any of the regions of interest. On average, the PEand the UM are ≈
600 K and ≈ ≈ ≈
50 deg moreinclined than that of the UM, which is ≈
20 deg with respect tothe line-of-sight.It might surprise the reader to see that the magnetic fieldlines of the UM are on average more inclined than one wouldexpect for this structure, where close-to-vertical fields are ex-pected. This is because the inclination values of the UM have abroad distribution from 0 to 35 deg, and only in the centremostpart of the UM is the magnetic field inclination really close tovertical fields. When moving from the very centre, the incli-nation increases, reaching values up to ≈
30 deg. That is why,when averaging the whole UM inclination, we get values closeto 20 deg.Analysing the temporal evolution of the di ff erent atmo-spheric parameters T , | B | , and γ of the three LBs, we found thatLB1 exhibits very little time variation, whereas LB2 and LB3do, LB2 heats and decreases its magnetic field strength, whileLB3 cools and increases its magnetic field strength, which be-comes more vertical. The temporal evolution of the parametersshows a variety of behaviours that could be related to the evo-lution process of the LB, as is seen in the HMI image (Figure2). The properties of LB3 behave as if they were changing froma penumbra-like structure into an umbra-like one. This could bedue, as is seen from HMI data, to the fact that this LB forms bythe coalescence of two UMs and finally disappears. Additionally,there are some di ff erences between LB1 and LB2. The tempera-ture of LB1 is closer to the penumbral temperature than that ofLB2. Also, the magnetic field strength of LB2 is closer to theUM magnetic fields than that of LB1, so LB1 essentially looksmore like the PE and LB2 behaves more like the UM. Table 6shows the quantitative di ff erences between the atmospheric pa-rameters of each LB and the PE, UM, and QS (when applicable)together with their standard deviation as a measurement of thedispersion.Analysing the global behaviour of the three LBs, it seemsthat there might be an anti-correlation between the temperatureand the magnetic field strength, meaning the cooler the LB, thegreater the magnetic field strength. Figure 10 shows this anti-correlation between T lb and | B | lb . The pink line represents thelinear fit, with linear and independent coe ffi cients − . G / kK and 6 . G , respectively, and the Pearson correlation coe ffi cientis − . Article number, page 10 of 27ame(s) of author(s): short title
Fig. 9.
From top to bottom: Temporal evolution of temperature, inclination, and magnetic field strength, respectively, averaged over the opticaldepth values of the left column of Tab. 5 and over the LB, UM, PE, and QS pixels. From left to right: LB1, LB2, and LB3, respectively. Black andgrey lines are for UM values, red and pink for LB pixels, and light and dark green for the PE, and, in the temperature panel, light and dark blue for the QS pixels. The diamonds correspond to the values and the dash-dotted lines indicate the 25th and 75th percentiles. The x -axis represents the scan, of which the format is day_scan-number. The previous analysis does not consider spatial properties of thephysical parameters in the LBs. In order to analyse this spatialbehaviour, Figures 11, 12, and 13 present the values of the at-mospheric parameters as a function of spatial location averagedover the optical depth listed in the left column of Tab. 5. Eachrow of the figures represents the analysed atmospheric parame-ter ( T , | B | , and γ ), and the di ff erent columns are the various ob- servations of each LB (the observation date and label are writtenat the top of the figure).The temporal evolution and the spatial distribution of LB1(Fig. 11) show that the LB area increases from a tiny intrusioninto the UM (Scan 2_02) until it breaks the UM into two di ff er-ent umbral cores (Scan 5_07). Some general properties observedin Fig. 9 are also seen here, such as the fact that T lb , | B | lb , and γ lb lie between those for the PE and the UM. Morphologically,the temperature in the LB shows a patchy structure where some Article number, page 11 of 27 & A proofs: manuscript no. aanda
Table 6.
Comparison of LB results with those of the PE, UM, and QS.
Region T lb − T reg | B | lb − | B | reg γ lb − γ reg (K) (G) (deg)UM 1460 ± − ±
120 7 ± − ±
97 450 ± − ± − ±
80 - -UM 1000 ± − ±
120 3 ± − ±
160 850 ± − ± − ±
160 - -UM 1240 ± − ±
120 13 ± − ±
150 520 ± − ± − ±
160 - -
Notes.
The first column is the LB number, the second is the region forthe comparison, and the last three columns correspond to the results for T , | B | , and γ , respectively. The last three columns were calculated asthe average of the di ff erences between the LB parameters and the regionparameters. The standard deviation of these averages is also shown aftereach di ff erence value. Fig. 10.
Relation between the magnetic field strength and the tempera-ture for the three LBs. The red triangles, blue squares, and green dia-monds correspond to the values for LB1, LB2, and LB3, respectively.The purple line represents the linear fit of the values whose equation iswritten on the right of the graph. The r value corresponds to the Pearsoncorrelation coe ffi cient. small structures are cooler than their surroundings (blue areas ofthe Scans 3_05 and 3_06). This cooler region could be relatedto the dark lane as it is more or less centred along the LB axis,but we lack good enough spatial resolution to assess this withconfidence. Also, some scans (3_03 and 3_04) show a suddenheating of the axis of the LB as compared to the rest of the LBbut the scarce temporal cadence does not allow us to further ad-dress this point. Regarding the magnetic field strength, values ofthe | B | lb are lower than those of the surrounding UM. It is worthnoting that the UM also presents a strong spatial variation as themagnetic field is weaker to the east of LB1 than to the west. Formost of the area, the magnetic field strength values are in therange ≈ Fig. 11.
Temporal evolution maps of the temperature, inclination in thelocal reference frame, and the magnetic field strength of LB1. The pa-rameters are averaged over the optical depths shown in the left columnof Tab. 5. The date and the hour of each observation are written at thetop of the figure. Arrows indicate the disc centre direction. than at the borders. Also, the magnetic field lines of LB1 show abroad inclination distribution from ≈ ≈ ff erent, probably because the heliocentric an-gle for this scan (see Tab. 1) is higher than for the rest observedmaps and this produces that the region of the atmosphere towhich the spectral lines are sensitive may be di ff erent (higher Article number, page 12 of 27ame(s) of author(s): short title
Fig. 12.
Same as Figure 11 for LB2. in the atmosphere) for this particular scan, showing a di ff erentbehaviour of inclination angle. All the magnetic LB structure isalmost vertical with values very similar to those of the UM. Aninteresting feature of the inclination in contrast to magnetic fieldstrength and temperature (which seem to be homogeneous alongthe whole LB) is that the magnetic lines are more vertical at thetip of LB1 (the end of the LB inside the UM) and more inclinedin the rest of the body. This property stays constant throughoutthe time sequence: when a LB is forming, T and | B | change theirvalues before γ does.The spatial evolution of LB2 (Figure 12) shows how the LBretreats until it almost disappears. A similar behaviour to thatof LB1 (Figure 11) is visible for this LB, but in this case themagnetic field strength of LB2 is higher than that of LB1 ( | B | lb ≈ | B | lb1 ≈ ≈ γ precedes T and | B | in its withdrawal.Light Bridge 3 (Figure 13) is formed when two UMs ap-proach each other and the remnant granulation gets trapped inbetween. In this LB, T and | B | behave much like those of LB1 Fig. 13.
Same as Figure 11 for LB3. and LB2 ( | B | lb3 < | B | um and T lb3 > T um ), except for the evidentpenumbral intrusion su ff ered by the LB in Scans 3_03, 3_04, and3_05. This event is characterised by a sudden change of | B | and γ , changing from 1750 G to 1500 G and 30 deg to 80 deg, re-spectively. This sudden change in LB properties is mentioned inSection 3.1.1, but now, with the spatial information, we specu-late that this event is caused by a sudden intrusion of penumbralfilament into the umbral area through the LB spine. Article number, page 13 of 27 & A proofs: manuscript no. aanda
Another way to study the behaviour of the LBs is to comparethe variation in optical depth of the atmospheric stratificationfor these structures with those seen for the PE and the UM. Todo this, we calculated the di ff erence between the average val-ues of various atmospheric parameters at higher optical depthsand the average at lower optical depths (the optical depth val-ues for which the various averages were made are shown in Tab.5, Di ff erences column) to obtain a rough sense of their verticalgradients. We did this calculation for T , | B | , and γ for the penum-bral, umbral, and LB regions. Figures 14 to 16 show these di ff er-ences and their probability density function (PDF) distributionfor LB1. We used it as a representative of all LBs as they arevery similar (see Appendix A for the results of LB2 and LB3).Having a broad distribution means that there is a large range ofgradients (steep and shallow), so the di ff erence between the pa-rameter value in the higher and lower layers can be large. Whenthe distribution is narrow, the whole set of pixels behave sim-ilarly. When it is centred on the value ’0’, this means that, onaverage, the analysed parameter does not change with height. Ifthe distribution is centred in the positive (negative) values, wesee that the values in the higher layers are, on average, larger(smaller) than in the lower ones.The penumbral area (green line) cools faster with opticaldepth than the UM, hence the gradient of temperature for thepenumbral area is lower than for the umbral one, as is seen inthe histograms of Figure 14. The temperature in the lower lay-ers for both regions is higher than in the upper layers, and theirdistributions have a similar widths ( ∆ T ≈ -1000 K), however, theaverage di ff erence for the UM is <∆ T > ≈ <∆ T > ≈ -2200 K. As for the magnetic field strength di ff erence(Figure 15), the umbral distribution has a width of ∆ | B | ≈
200 Gand its average is <∆ | B | > ≈ −
200 G (the negative sign meansthat at higher layers the magnetic field strength is weaker), whilethe penumbral distribution is broader ( ∆ | B | ≈
400 G) and themean value is <∆ | B | > ≈ −
450 G. A similar behaviour is seenfor the magnetic field inclination di ff erence distribution (see Fig-ure 16). For the umbral region, the histogram is narrower (itswidth is ∆ γ ≈ ∆ γ ≈
19 deg), and the average umbral distribution is <∆ γ > ≈ -0.25 deg and <∆ γ > ≈ -1.5 deg for the UM and PE, respectively(i.e. the orientation of the magnetic field lines hardly changeswith optical depth for the UM and PE).Focusing on the LBs, the distribution of the di ff erences intemperature (see Figures 14 and the figures in Appendices A,A.1, and A.2, the last two figures for LB2 and LB3) shares sim-ilarities with the UM. The distributions are broad with a widthof ∆ T ≈
800 K and the average is <∆ T > ≈ − ff erences of the LB are closer to the mini-mum of the umbral distribution (around − ∆ T ≈ − ff erence maps for the di ff erent scans of the LBs. They showtwo behaviours: the borders of the LBs cool less than some ar-eas in the middle of the structure, which cool faster with opticaldepth. This behaviour could be related to the dark lane reportedby other authors (Sobotka et al. 1994, Bharti et al. 2007, Louiset al. 2008, Rouppe van der Voort et al. 2010, see Section 1 for Fig. 14.
Distribution of the di ff erence between the average temperaturein the upper and lower layers of LB1 ( ∆ T = T upper − T lower ). The red linecorresponds to the LB distribution. The penumbral is shown in greenand the umbral distribution in black. Fig. 15.
Same as Figure 14, but for the magnetic field strength of LB1( ∆ | B | = | B | upper − | B | lower ). more details), which we are unable to see at this spatial resolu-tion.The global behaviour of the magnetic field strength for all thescans of the LBs (Fig. 15 and figures from Appendix A, A.5, andA.6) shows a trend towards negative values, that is, the magneticfield strength of deeper layers is stronger than that in the upperones. The width of the distributions is ∆ | B | ≈
500 G, similar tothat of the PE.The distribution of the magnetic field inclination (Fig. 16 andfigures from Appendices A, A.7, and A.8) is broad (its width is ∆ γ ≈
11 deg), and the average is <∆ γ > ≈ ff erently with respect to boththe PE and the UM as its distribution peaks to positive (in mostof them) ∆ γ (more horizontal above), and its shape is broaderthan that for the UM and narrower than that inferred for the PE. f f , v m los , and v nmlos Other important parameters to analyse are the filling factor ( f f )and the line-of-sight velocity ( v los ). Because of the strategy fol- Article number, page 14 of 27ame(s) of author(s): short title
Fig. 16.
The same as Figure 14, but for the magnetic field inclination ofLB1 ( ∆ γ = γ upper − γ lower ). Fig. 17. Di ff erence maps between the average temperature in the upperand lower layers of LB1. lowed in the Stokes parameter inversion for the LBs, we havea line-of-sight velocity of the magnetic component, v m los , and aline-of-sight velocity of the non-magnetic atmosphere v nmlos . First,we analysed the behaviour of the f f values, and then we consid-ered two di ff erent analyses: 1) a comparison of the v mlos with v nmlos , and 2) a characterisation of the variation in the optical depth of v mlos ( v nmlos is constant with optical depth). The latter is also donefor the PE and UM. In order to infer the atmospheric parameters of the observedspectral profiles, we inverted the Stokes parameters using an in-version strategy (see Section 2.4) with two atmospheric com-ponents: one magnetic and one non-magnetic, weighted by the f f . This way, the f f goes with the magnetic component, while(1 − ff ) goes with the non-magnetic.Table 7 shows the median of the f f for each scan and LBtogether with two percentiles (14 th and 86 th , sub- and super- in-dex, respectively). For LB1, the thickest one, there are clearlytwo di ff erent behaviours (see i.e. top-right panel of Figures 18and B.1): the core is highly non-magnetic, while for the outer parts of the LB the magnetic f f increases. For LB2 and LB3,this behaviour is not that clear, which may be related to the factthat these LBs are thinner (figures from Appendices B.2 and B.3)and they are mainly magnetic. Fig. 18.
Comparison between magnetic and non-magnetic depth-averaged velocities of LB1 in Scan 3_05. The histogram shows the dis-tributions of v mlos (black line) and v nmlos (red line). The upper right panelcorresponds to f f , and the white asterisk indicates the pixel taken toanalyse the Stokes I and V profiles in Fig. B.2. The lower panels repre-sent v mlos on the left and v nmlos on the right. The black contours of the mapsenclose the area with f f below 85%. v m los and v nmlos We now focus on the behaviour of the velocities of the mag-netic ( v mlos ) and non-magnetic ( v nmlos ) components on the LB. Todo so, we compared both velocities by analysing their veloc-ity distributions averaged between the optical depths shown inTab. 5 (Sensitivity Range column). This is not relevant for the v nmlos case as it is considered to be constant with optical depth.For the non-magnetic component, we avoided those LB pixelswith f f above 75% because, in most of these cases, this compo-nent was used to fit non-spectral line features (see Section 2.4).Figure 18 shows this comparison for LB1 in Scan 3_05 (the fig-ures for the other scans and LBs are in Appendix B, from B.1to B.17, as similar results are observed for most of them). Thetop-left panel of the figure shows the distribution of v mlos and v nmlos .The magnetic distribution (black line) is narrow with a width of ∆ v mlos ≈ / s centred close to 0 km / s, so the magnetic compo-nent for LB1 is essentially at rest, while the distribution for v nmlos (red line) is broader (its width is ∆ v nmlos ≈ / s) with values upto 4 km / s and with an average value of 2 km / s. This result is thesame for all the scans and LBs except for the one shown in B.1.A possible explanation for a di ff erent result can be seen in B.1.The top-right panel of Figure 18 shows the f f map. Thisparameter shows a clear spatial distribution, which is smallerclose to the centre of the LB (less than ≈ v nmlos map (bottom right panel of Fig- Except for the LB1 in Scan 2_02 shown in Figure B.1. A possibleexplanation for this di ff ference is given in Appendix B.Article number, page 15 of 27 & A proofs: manuscript no. aanda
Table 7.
Median of the filling factor for each scan and LB.
LB Scan2_01 2_02 3_03 3_04 3_05 3_06 5_07LB1 - 0 . . . . . . . . . . . . . . . . . . LB2 - 0 . . . - 0 . . . . . . . . . -LB3 0 . . . . . . . . . . . . . . . . . . -ure 18). The spine has areas of zero velocities or even upflowsthat are surrounded by (strong) downflows. The areas where therest / upflows are present coincide with the area with smaller f f values. This means that in those areas the LBs are mainly non-magnetic and show a convective-like pattern. In clear contrast,the magnetic component shows an average downflow character.A similar behaviour is seen in all the other scans and LBs exceptScan 2_02 (see Appendix A for the results and for a discussionof this exception). v mlos optical depth difference characterisation Following the same method used in Section 3.1 for T , | B | , and γ ,we calculated the di ff erences of v mlos between the averaged upperlayers and the averaged lower layers (the optical depth values toaverage are shown in the Tab. 5, Di ff erences column). Figures19, 20, and 21 show the histograms of this analysis for LB1,LB2, and LB3, respectively. Fig. 19.
Same as Figure 14, but for the line-of-sight velocity of LB1 ofthe magnetic component ( ∆ v los = v upperlos − v lowerlos ). The distribution of the penumbral velocity di ff erences forall the scans (i.e. Figure 19, in green) is narrow and the meanis < v mlos > ≈ − .
05 km / s. Also, the distributions have long tailsreaching up to ± / s . This di ff erence for the UM is charac-terised by positive values and an asymmetric distribution. Themean value of this distribution is around 0.5 km / s, with valuesof 0.15 km / s of averaged velocities in the upper layers and -0.25 km / s in the lower layers, meaning there are weak averagedownflows in higher layers and weak average upflows in lowerones. On average, both distributions have a width of ≈ / s. Fig. 20.
Same as Figure 19, but for LB2.
Fig. 21.
Same as Figure 19, but for LB3.
For LBs, there are two di ff erent behaviours, as seen fromthe distributions of the v mlos di ff erences between upper and lowerlayers. LB1 and LB3 are very similar with average values of ≈ -0.15 km / s and their width being ≈ / s. This is similar to thepenumbral case, but the distributions of the di ff erences for theLBs are slightly shifted to negative values. As for LB2, the dis-tribution width is ≈ / s, and its values tend to be positivewith an average of ≈ / s. The di ff erence between the be-haviours of LB1 and LB3 and that of LB2 could be because LB2is very thin and there could be contamination of the surroundingumbra, as LB2’s behaviour is more similar to the umbral one. Article number, page 16 of 27ame(s) of author(s): short title
4. Discussion and conclusions
Spectropolarimetric data observed with GRIS have been used tostudy the long-term evolution of the thermodynamic and mag-netic parameters of three LBs formed in the same sunspot. Thespectropolarimetric data were inverted in order to infer the atmo-spheric parameters and study the temporal evolution (spanning afew days), spatial distribution, and optical depth variation of dif-ferent atmospheric parameters in a sunspot with 3 LBs: the tem-perature ( T ), the magnetic field strength ( | B | ), the local magneticfield inclination ( γ ), and the line-of-sight velocity ( v los ) in threeLBs hosted by the same sunspot. Also, the relevance of the mag-netic field was studied through the analysis of the relative fillingfactor ( f f ). Using the continuum intensity provided by the HMIinstrument on board the SDO satellite, we find that each LB wasformed in a di ff erent way. LB1 was formed by a strong intrusionof the PE that grows until it breaks the UM into two parts, LB2formed by a weak intrusion of the PE that does not break theUM completely and then retreats until it disappears, and LB3was formed by the remnant granulation as two UMs approacheach other and (eventually) merge. Data from the GRIS instru-ment allowed us to study the topology and thermodynamics ofthese LBs in more detail.In general, the three LBs are characterised by similar atmo-spheric structures and behaviours. In order to explain the ob-served spectral profiles of LBs, we need two di ff erent atmo-spheric components in the inversion process: one magnetic andone non-magnetic. The magnetic component occupies a signif-icant fraction of LB pixels, reaching up to 80% at the borders,while in the axis of the LB it occupies between 10 and 60%. Re-garding the physical parameters, these structures are hotter thanUM and cooler than PE and the QS. Their magnetic field strengthis lower than that of the UM and higher than the magnetic fieldof the PE. Furthermore, the magnetic field lines of LBs are moreinclined than the magnetic lines of the UM and more verticalthan those of the PE.Analysing the temporal evolution of the di ff erent atmo-spheric parameters, we find that LB1 exhibits very little timevariation, whereas LB2 and LB3 exhibit much more. LB2 heatsand decreases its magnetic field strength, while LB3 cools andincreases its magnetic field strength, which becomes more verti-cal. Furthermore, it looks as if the properties of LB3 are chang-ing from a penumbra-like structure into an umbra-like one. Thiscould be due, as is seen from HMI data (Figure 2), to the fact thatthis LB forms by the coalescence of two UMs and finally disap-pears. Additionally, there are some di ff erences between LB1 andLB2. The temperature of LB1 is closer to the penumbral temper-ature than that of LB2. Also, the magnetic field strength of LB2is closer to the UM magnetic fields than that of LB1, so LB1 es-sentially looks more like the PE and LB2 behaves more like theUM.This general picture is in agreement with the bibliography(see Tab. 8 for a summary of quantitative measurements of LBsin the literature). We find average temperature values of 6000 K,5600 K, and 5900 K for LB1, LB2, and LB3, respectively. Thesevalues are slightly hotter than those found by Jurˇcák et al. (2006)and Lagg et al. (2014). This could be due to the fact that weused observations in the near infrared, while they observed inthe visible, so we probed deeper (hotter) into the atmosphere.We note that the temperature value in Falco et al. (2016) is anupper boundary (for the biggest granules of the LB).On average, the magnetic field strengths found for LB1, LB2,and LB3 are 1800 G, 2300 G, and 1900 G, respectively. Thesevalues are similar to those found by Kneer (1973), Rüedi et al. (1995), Leka (1997), Jurˇcák et al. (2006), Rezaei et al. (2012),Shimizu (2011), Louis et al. (2012), Louis et al. (2014), Louiset al. (2015), and Falco et al. (2016). Our data suggest that theremight be an anti-correlation between the temperature and themagnetic field strength: the cooler the LBs, the stronger its mag-netic field.The magnetic field lines of the LBs are more horizontal thanthose of the UM. The averaged values for LB1, LB2, and LB3are 25, 20, and 30 deg, respectively. These values are in agree-ment with those reported by Leka (1997), Jurˇcák et al. (2006),Louis et al. (2012), Louis et al. (2014), Louis et al. (2015), andGuglielmino et al. (2019). As before, despite the similar globalbehaviour, LB3 seems to show an evolution with time of theaverage inclination. Initially, the magnetic field lines are quiteinclined ( ≈
40 deg), and, as time evolves, they shift to an aver-age of ≈
25 deg. In addition, on May 3, LB3 su ff ers an abruptchange to more inclined fields, which is in agreement with theHMI sequence (a protrusion of the PE into the UM is seen in thesouthern part of the LB in the panel ‘20:00 02 / May / T , γ, and | B | of the LBs is in gen-eral agreement with previous results of other authors ( | B | um > | B | lb > | B | pe , T pe > T lb > T um , and γ pe > γ lb > γ um ). In addition,there are some features, such as the intrusion that LB3 su ff ersin Scans 3_03, 3_05, and 3_06, where the physical propertiesare a bit di ff erent than for the rest of the LB (almost horizon-tal magnetic field lines, weak magnetic field strength, and hottertemperatures). Moreover, we found that the spatial extension ofthe LB-like physical properties is di ff erent for T lb and | B | lb thanfor γ lb , as the former extend more than the latter. The inclinationat the tip of LBs is always more vertical (similar to the umbralvalues) than the rest of the LB. This result could be consistentwith the work done by Lindner et al. (2020), who studied theJurˇcák criterion (Jurˇcák et al. 2018). This is probably a stabil-ity criterion for the UM (though this is still an open discussion,see Löptien et al. 2020). For this particular sunspot, they foundthat the Jurˇcák criterion applies to part of the LB, meaning itcould be related to the part of the LB that has higher inclina-tions, while the tip would be ascribed to the fact that it is ‘underevolution / formation’. Another possibility is that the LB tip is as-sociated with the emergence of a horizontal magnetic field (To-riumi et al. 2015a), but we do not observe the convection patternin the magnetic velocity so we would need more information toconfirm if this is due to the horizontal emergence flux. Also, thisstudy reveals another interesting feature: the biggest LB is wideenough to show a cool and weak patchy structure along the axisof the LB. This result could imply the presence of a dark lanewhose existence cannot be determined as the spatial resolutionis not high enough.Making a di ff erential analysis between two optical depths wefound that the PE cools the most (2200 K) with optical depth,while the UM and LB are quite similar, with the latter coolingdown slightly less than the UM. This result for the LB tempera-ture is consistent with previous studies (see i.e. Jurˇcák et al. 2006and Lagg et al. 2014 from Tab. 8). Moreover, the temperature ofthe LBs shows an additional peak ( ∆ T ≈ -3000 K) for some scanswhere the temperature changes faster with optical depth. These Article number, page 17 of 27 & A proofs: manuscript no. aanda
Table 8.
Comparison of the results presented in this chapter with previous studies.
Paper Spectral Lines (Å) Region | B | (G) γ (deg) T (K)MIN MAX < > MIN MAX < > < >
Kneer (1973) 6150 LB Umbra LB sc LB wc LBs
Umbra LB narrow km LB narrow km LB narrow km LB broad km
705 44 5990 LB broad km
860 39 5310 LB broad km LB km LB km LB km Umbra km Umbra km Umbra km Umbra km Umbra km Umbra km LB Umbra LB LB LB LB LB RF LB Intlog ( τ ) =
170 108 6590 LB Intlog ( τ ) = − .
60 108 5330 LB Intlog ( τ ) = −
280 120 4810 LB Boulog ( τ ) =
320 95 6290 LB Boulog ( τ ) = − . LB Boulog ( τ ) = − LB
800 1700 140 180Louis et al. (2015) 6301.5 & 6302.5 LB Umbra LB N
700 1000 6400* LB S LB log ( τ ) = .
30 76 LB log ( τ ) = − . LB UF Umbra
Penumbra LB LB LB Umbra
Notes.
The first column is the paper reference, the second is the spectral lines used for the study, the third is the region analysed, the next sixcolumns correspond to the minimum, maximum, and mean of the magnetic field strength and inclination, and the last column is the averagetemperature. When the authors do not study a parameter, the cell is empty. The meaning of the subindices are: sc and wc from Rüedi et al. 1995 arestrong and weak atmospheric components, respectively; FR from Louis et al. 2012 means LB in formation; N and S from Falco et al. 2016 meansthe north and south area of the LB; and * from Falco et al. 2016 means that the temperature was calculated in the biggest granule of the northernpart of the LB. The superindices from Lagg et al. 2014 mean that the values are calculated in the interior and in the boundary of a granule in a LB.Article number, page 18 of 27ame(s) of author(s): short title secondary peak values are located along the axis of the LB andcould be related to the dark lane reported by other authors. Themagnetic field strengths of the UM, PE, and LB are stronger inthe lower layers than in the upper ones. The PE shows the steep-est decrease in magnetic field, the UM the shallowest, and theLB magnetic field decreases between these two layers. Regard-ing the inclination of the magnetic field lines, the UM shows noaverage change between the two layers considered, while the PEand LBs do show an average variation of inclination with opti-cal depth, with more inclined field lines in the upper layers forthe LBs and with more vertical field lines in the upper layers forthe PE. The results for the magnetic parameters are opposite tothose presented by Jurˇcák et al. (2006) and Lagg et al. (2014),as in our case | B | decreases and becomes more horizontal withoptical depth. Whether this di ff erence arises from the di ff erentspectral range used or because the LBs are intrinsically di ff er-ent from those studied by these authors is unclear. Ideally, LBsobserved simultaneously in both wavelength ranges would be re-quired to further address this point.Analysing the filling factor values shown in Tab. 7, we cansee that the cores of LBs are highly non-magnetic, while theouter parts have a magnetic character. The fact that between 6%and 29% (as inferred from the median value of the LBs analysedon this paper) of the LBs are non-magnetic is in agreement withLeka (1997), who observed that often more than 20% of the ma-terial in the LB is non-magnetic. An open question that remainsfrom this work concerning the filling factor is the fact that theouter parts of the LBs are much more magnetic. This could berelated to 1) a residual of stray light that could not be removedusing the techniques applied to the data, or 2) the optical depthsthe observed spectral line is sensitive to are di ff erent for the coreand the outer parts of the LB, that is, the physical propertiesprobed by the spectral line are di ff erent in the core and the outerparts of the LBs.The analysis of v mlos shows that the UM is almost at rest, whilethe velocity di ff erence of the penumbral material is positive withoptical depth. We find that LB1 and LB3 behave similarly andhardly change with optical depth, while LB2 behaves like thePE with a positive velocity di ff erence with optical depth (thevelocity in the upper layers being 0.2 km / s greater than in thelower layers). v nmlos shows a common behaviour for all the LBsand scans (save for one, whose behaviour we cannot explain).We found that the velocity of the magnetic component is closeto zero or shows small downflows, and the velocity of the non-magnetic component has a convective-like character. The spineof the LBs is at rest or even has small upflows and is surroundedby strong downflows. These results are in agreement with thebibliography (see Tab. 9 for a summary of v los measurements ofLBs in the literature). Some authors, including Rimmele (2008),Giordano et al. (2008), Rouppe van der Voort et al. (2010), Lagget al. (2014), Toriumi et al. (2015b), and Guglielmino et al.(2017), found that the spine of the LBs shows upflows, whilethe edges show downflows (the same behaviour as we see in thenon-magnetic atmosphere). However, they do not distinguish be-tween magnetic and non-magnetic components, so it is not easyto compare directly. Also, the v mlos values calculated in this pa-per are similar to the results obtained by Rüedi et al. (1995) andShimizu (2011), who observed downflows in the whole LB. Acknowledgements.
ABGM would like to thank Matthias Rempel for the revi-sion of the paper and his suggestions. The authors are grateful to the SDO / HMIteam for their data. ABGM acknowledges Fundación La Caixa for the finan-cial support received in the form of a Ph.D. contract. This work was supportedby NASA Contract NAS5-02139 (HMI) to Stanford University. We thank theUniversity Corporation for Atmospheric Research and their NCAR / HAO pro- grams. This work was supported by the HAO Newkirk Fellowship. This projecthas received funding from the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme (SUNMAG,grant agreement 759548). The Institute for Solar Physics is supported by a grantfor research infrastructures of national importance from the Swedish ResearchCouncil (registration number 2017-00625). This material is based upon worksupported by the National Center for Atmospheric Research, which is a majorfacility sponsored by the National Science Foundation under Cooperative Agree-ment No. 1852977. The authors gratefully acknowledge support from the Span-ish Ministry of Economy and Competitivity through project AYA2014-60476-P(Solar Magnetometry in the Era of Large Solar Telescopes),as well as projectPGC2018-102108-B-I00 and FEDER funds. This paper made use of the IAC Su-percomputing facility HTCondor (http: // research.cs.wisc.edu / htcondor / ), partlyfinanced by the Ministry of Economy and Competitiveness with FEDER funds,code IACA13-3E-2493. References
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Table 9.
Comparison of the results of the v los obtained in this work with previous studies. Paper Spectral Lines Upflows Downflows Upflows & Downflows(Å) (km / s) (km / s) (km / s)Beckers & Schröter (1969) 6173.3 & 5576.1 URüedi et al. (1995) 15648.5 & 15652.9 1.5Leka (1997) 6301.5 & 6302.5 U & DRimmele (1997) 5576 & 5691 U & DHirzberger et al. (2002) 5425 0.2 - 1.5Schleicher et al. (2003) 15648.5 & 15652.9 0.3Bharti et al. (2007) 5576 & 6302 U: 0.1 - 0.2 on the LB axisD: 0.5 - 1.4 on the LB edgesKatsukawa et al. (2007) 6301.5 & 6302.5 0.2Rimmele (2008) 5434 UGiordano et al. (2008) 7090.4 U: 0.07 on the LB axisD: 0.15 on the LB edgesLouis et al. (2009) 6301.5 & 6302.5 SupersonicRouppe van der Voort et al. (2010) 6301.5 & 6302.5 U: 0.5 - 1 on the LB axisD: lower on the LB edgesShimizu (2011) 6301.5 & 6302.5 0.5 - 2Lagg et al. (2014) 6301.5 & 6302.5 U: 2 on the LB axisD: supersonic on the LB edgesToriumi et al. (2015b) 6301.5 & 6302.5 U: 1 on the LB axisD: 6 on the LB edgesGuglielmino et al. (2019) 6301.5 & 6302.5 DThis work 15648.5 & 15662.0 U: on the LB axisD: on the LB edges Notes.
The first column is the reference paper, the second is the spectral lines used for the study, and the third, fourth, and fifth columns show ifthe LB shows upflows, downflows or both and their values, respectively. U means upflows and D downflows.
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Article number, page 20 of 27ame(s) of author(s): short title
Appendix A: Height difference characterisation ofthe magnetic and thermal parameters | B | , γ , and T In this appendix, we present the results of the LBs described inSection 3.1. All figures of this appendix show the height di ff er-ence characterisation. Figures A.1–A.4 are for the temperatureanalysis, Figures A.5 and A.6 are for the magnetic field strengthanalysis, and Figures A.7 and A.8 for the inclination analysis.The black line is for the pixels of the umbra, the green one is forthe pixels from the penumbra, and red represents the distributionof the LB pixels. Appendix A.1: Temperature
Fig. A.1.
Same as Figure 14, but for LB2.
Fig. A.2.
Same as Figure 14, but for LB3.
Fig. A.3.
Same as Figure 17, but for LB2.
Fig. A.4.
Same as Figure 17, but for LB3.Article number, page 21 of 27 & A proofs: manuscript no. aanda
Appendix A.2: Magnetic field strength
Fig. A.5.
Same as Figure 15, but for LB2.
Fig. A.6.
Same as Figure 15, but for LB3.
Appendix A.3: Magnetic field inclination
Fig. A.7.
Same as Figure 16, but for LB2.
Fig. A.8.
Same as Figure 16, but for LB3.Article number, page 22 of 27ame(s) of author(s): short title
Appendix B: Comparison between v m los and v nmlos In this appendix, we present the comparison between the line-of-sight velocities of the magnetic and non-magnetic componentsfor the three LBs in the di ff erent scans. These plots were calcu-lated following the procedures explained in Section 3.2.2. Appendix B.1: Light Bridge 1
Fig. B.1.
Same as Figure 18, but for LB1 of Scan 2_02.
Figure B.1 shows the comparison of v mlos and v nmlos for LB1 inScan 2_02. The histogram shows that the distribution of v mlos issimilar to that of Figure 18, narrow and around 0 km / s, whilethe distribution of v nmlos has negative values (instead of positive).In order to find out whether this change in behaviour is the re-sult of the data reduction, deconvolution or inversion process, orwhether there is an error in the data analysis, we compared theStokes I and V profiles of two di ff erent pixels (white asterisksof Figures 18 and B.1). These pixels were selected because bothatmospheric components have a similar contribution to the finalprofile ( f f ≈ I profiles and compared it withthe mid-point of the Stokes V profiles (see Figure B.2). For Scan3_05, the minimum of the Stokes I (orange line of top panel) isredshifted with respect to the mid-point of Stokes V (purple lineof bottom panel), while the minimum of the Stokes I (dark blueline of top panel) of Scan 2_02 is blueshifted with respect to themid-point of Stokes V (green line of bottom panel). Thus, theprofiles are consistent with the results obtained with SIR. Now,the reason why the v nmlos distributions for the three LBs for this sin-gle scan are blueshifted is not clear and needs additional study.We can be sure that it is not a problem of the velocity calibrationsince the UM, PE, and QS behave consistently. Fig. B.2.
Comparison between Stokes I and V for the pixels marked bywhite asterisks in Figures 18 and B.1. The red and black lines corre-spond to the Stokes I and V profiles of Scans 3_05 and 2_02, respec-tively. The vertical solid lines of the top panel and the dashed lines ofthe bottom panel represent the core of the Stokes I profiles. The verticallight blue and pink lines of the bottom panel mark the red and blue lobesof the Stokes V profiles for Scans 2_02 and 3_05, respectively, and thegreen and purple lines correspond to the mid-point of Stokes V profiles. Fig. B.3.
Same as Figure 18, but for LB1 of Scan 3_03.Article number, page 23 of 27 & A proofs: manuscript no. aanda
Fig. B.4.
Same as Figure 18, but for LB1 of Scan 3_04.
Fig. B.5.
Same as Figure 18.
Fig. B.6.
Same as Figure 18, but for LB1 of Scan 3_06.
Fig. B.7.
Same as Figure 18, but for LB1 of Scan 5_07.Article number, page 24 of 27ame(s) of author(s): short title
Appendix B.2: Light Bridge 2
Fig. B.8.
Same as Figure 18, but for LB2 of Scan 2_02.
Fig. B.9.
Same as Figure 18, but for LB2 of Scan 3_04.
Fig. B.10.
Same as Figure 18, but for LB2 of Scan 3_05.
Fig. B.11.
Same as Figure 18, but for LB2 of Scan 3_06.Article number, page 25 of 27 & A proofs: manuscript no. aanda
Appendix B.3: Light Bridge 3
Fig. B.12.
Same as Figure 18, but for LB3 of Scan 2_01.
Fig. B.13.
Same as Figure 18, but for LB3 of Scan 2_02.
Fig. B.14.
Same as Figure 18, but for LB3 of Scan 3_03.
Fig. B.15.
Same as Figure 18, but for LB3 of Scan 3_04.Article number, page 26 of 27ame(s) of author(s): short title
Fig. B.16.
Same as Figure 18, but for LB3 of Scan 3_05.