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At least 145 records · Page 8

Synergy of Observations and Dynamo Models to Understand and Predict Solar Activity Cycles

The long-standing problem of understanding the evolution of the global magnetic fields that drive solar activity through different temporal scales is becoming more tractable because, in addition to 400 years of sunspot records, we now have almost 4 solar cycles of magnetic field observations. These observations allow us to discern physical connections between dynamo model variables and observations using data assimilation analysis. In particular, the Ensemble Kalman Filter approach takes into account uncertainties in both observations and modeling and allows us to make reliable forecasts of solar cycle activity by using a relatively simple non-linear dynamical model of the solar dynamo. To expand this approach for more complex 2D and 3D dynamo modeling, it is necessary to decompose the observed synoptic magnetograms into poloidal and toroidal field components. In this presentation I will present initial results on magnetogram decomposition and assimilation of magnetogram data into dynamo modeling.

Kitiashvili, Irina N.↗

Periodicity of the solar full-disk magnetic fields

A full-disk solar magnetogram has been measured each day since 1970 January 19, and the daily Magnetic Plage Strength Index (MPSI) and the daily Mount Wilson Sunspot Index (MWSI) were calculated for each magnetogram at the Mount Wilson Observatory. The MPSI and MWSI are used to investigate the periodicity of the solar full-disk magnetic activity through autocorrelation analyses. Just two periods, the solar cycle and the rotation cycle, are determined in both the MPSI (the solar full-disk weak magnetic field activity) and MWSI (the solar full-disk strong magnetic field activity) with no annual signal found. The solar cycle for MPSI (10.83 yr) is found to be obviously longer than that for MWSI (9.77 yr). The rotation cycle is determined to be 26.8 ± 0.63 sidereal days for MPSI and 27.4 ± 2.4 sidereal days for MWSI. The rotation cycle length for MPSI is found to fluctuate around 27 days within a very small amplitude, but for MWSI it obviously temporally varies with a rather large amplitude. The rotation cycle for MWSI seems longer near solar minimum than at solar maximum. Cross-correlation analyses of daily MPSI and MWSI are carried out, and it is inferred that the MPSI components partly come from relatively early MWSI measurements.

79 ASTRONOMY AND ASTROPHYSICS↗

A NEW IMPLEMENTATION OF THE MAGNETOHYDRODYNAMICS-RELAXATION METHOD FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION IN THE SOLAR CORONA

The magnetic field in the solar corona is usually extrapolated from a photospheric vector magnetogram using a nonlinear force-free field (NLFFF) model. NLFFF extrapolation needs considerable effort to be devoted to its numerical realization. In this paper, we present a new implementation of the magnetohydrodynamics (MHD) relaxation method for NLFFF extrapolation. The magnetofrictional approach, which is introduced for speeding the relaxation of the MHD system, is realized for the first time by the spacetime conservation-element and solution-element scheme. A magnetic field splitting method is used to further improve the computational accuracy. The bottom boundary condition is prescribed by incrementally changing the transverse field to match the magnetogram, and all other artificial boundaries of the computational box are simply fixed. We examine the code using two types of NLFFF benchmark tests, the Low and Lou semi-analytic force-free solutions and a more realistic solar-like case constructed by van Ballegooijen et al. The results show that our implementation is successful and versatile for extrapolations of either the relatively simple cases or the rather complex cases that need significant rebuilding of the magnetic topology, e.g., a flux rope. We also compute a suite of metrics to quantitatively analyze the results and demonstrate that the performance of our code in extrapolation accuracy basically reaches the same level of the present best-performing code, i.e., that developed by Wiegelmann.

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THE LIMIT OF MAGNETIC-SHEAR ENERGY IN SOLAR ACTIVE REGIONS

It has been found previously, by measuring from active-region magnetograms a proxy of the free energy in the active region's magnetic field, (1) that there is a sharp upper limit to the free energy the field can hold that increases with the amount of magnetic field in the active region, the active region's magnetic flux content, and (2) that most active regions are near this limit when their field explodes in a coronal mass ejection/flare eruption. That is, explosive active regions are concentrated in a main-sequence path bordering the free-energy-limit line in (flux content, free-energy proxy) phase space. Here, we present evidence that specifies the underlying magnetic condition that gives rise to the free-energy limit and the accompanying main sequence of explosive active regions. Using a suitable free-energy proxy measured from vector magnetograms of 44 active regions, we find evidence that (1) in active regions at and near their free-energy limit, the ratio of magnetic-shear free energy to the non-free magnetic energy the potential field would have is of the order of one in the core field, the field rooted along the neutral line, and (2) this ratio is progressively less in active regions progressively farther below their free-energy limit. Evidently, most active regions in which this core-field energy ratio is much less than one cannot be triggered to explode; as this ratio approaches one, most active regions become capable of exploding; and when this ratio is one, most active regions are compelled to explode.

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PREPROCESSING MAGNETIC FIELDS WITH CHROMOSPHERIC LONGITUDINAL FIELDS

Nonlinear force-free field (NLFFF) extrapolation is a powerful tool for the modeling of the magnetic field in the solar corona. However, since the photospheric magnetic field does not in general satisfy the force-free condition, some kind of processing is required to assimilate data into the model. In this paper, we report the results of new preprocessing for the NLFFF extrapolation. Through this preprocessing, we expect to obtain magnetic field data similar to those in the chromosphere. In our preprocessing, we add a new term concerning chromospheric longitudinal fields into the optimization function proposed by Wiegelmann et al. We perform a parameter survey of six free parameters to find minimum force- and torque-freeness with the simulated-annealing method. Analyzed data are a photospheric vector magnetogram of AR 10953 observed with the Hinode spectropolarimeter and a chromospheric longitudinal magnetogram observed with SOLIS spectropolarimeter. It is found that some preprocessed fields show the smallest force- and torque-freeness and are very similar to the chromospheric longitudinal fields. On the other hand, other preprocessed fields show noisy maps, although the force- and torque-freeness are of the same order. By analyzing preprocessed noisy maps in the wave number space, we found that small and large wave number components balance out on the force-free index. We also discuss our iteration limit of the simulated-annealing method and magnetic structure broadening in the chromosphere.

79 ASTRONOMY AND ASTROPHYSICS↗

GUIDING NONLINEAR FORCE-FREE MODELING USING CORONAL OBSERVATIONS: FIRST RESULTS USING A QUASI-GRAD-RUBIN SCHEME

At present, many models of the coronal magnetic field rely on photospheric vector magnetograms, but these data have been shown to be problematic as the sole boundary information for nonlinear force-free field extrapolations. Magnetic fields in the corona manifest themselves in high-energy images (X-rays and EUV) in the shapes of coronal loops, providing an additional constraint that is not at present used as constraints in the computational domain, directly influencing the evolution of the model. This is in part due to the mathematical complications of incorporating such input into numerical models. Projection effects, confusion due to overlapping loops (the coronal plasma is optically thin), and the limited number of usable loops further complicate the use of information from coronal images. We develop and test a new algorithm to use images of coronal loops in the modeling of the solar coronal magnetic field. We first fit projected field lines with those of constant-{alpha} force-free fields to approximate the three-dimensional distribution of currents in the corona along a sparse set of trajectories. We then apply a Grad-Rubin-like iterative technique, which uses these trajectories as volume constraints on the values of {alpha}, to obtain a volume-filling nonlinear force-free model of the magnetic field, modifying a code and method presented by Wheatland. We thoroughly test the technique on known analytical and solar-like model magnetic fields previously used for comparing different extrapolation techniques and compare the results with those obtained by currently available methods relying only on the photospheric data. We conclude that we have developed a functioning method of modeling the coronal magnetic field by combining the line-of-sight component of the photospheric magnetic field with information from coronal images. Whereas we focus on the use of coronal loop information in combination with line-of-sight magnetograms, the method is readily extended to incorporate vector-magnetic data over any part of the photospheric boundary.

79 ASTRONOMY AND ASTROPHYSICS↗

LONG-TERM MEASUREMENTS OF SUNSPOT MAGNETIC TILT ANGLES

Tilt angles of close to 30,600 sunspots are determined using Mount Wilson daily averaged magnetograms taken from 1974 to 2012, and SOHO/MDI magnetograms taken from 1996 to 2010. Within a cycle, more than 90% of sunspots have a normal polarity alignment along the east-west direction following Hale's law. The median tilts increase with increasing latitude (Joy's law) at a rate of {approx}0.{sup 0}5 per degree of latitude. Tilt angles of spots appear largely invariant with respect to time at a given latitude, but they decrease by {approx}0.{sup 0}9 per year on average, a trend that largely reflects Joy's law following the butterfly diagram. We find an asymmetry between the hemispheres in the mean tilt angles. On average, the tilts are greater in the Southern than in the Northern Hemisphere for all latitude zones, and the differences increase with increasing latitude.

79 ASTRONOMY AND ASTROPHYSICS↗

NON-THERMAL RESPONSE OF THE CORONA TO THE MAGNETIC FLUX DISPERSAL IN THE PHOTOSPHERE OF A DECAYING ACTIVE REGION

We analyzed Solar Dynamics Observatory line-of-sight magnetograms for a decaying NOAA active region (AR) 11451 along with co-temporal Extreme-Ultraviolet Imaging Spectrometer (EIS) data from the Hinode spacecraft. The photosphere was studied via time variations of the turbulent magnetic diffusivity coefficient, {eta}(t), and the magnetic power spectrum index, {alpha}, through analysis of magnetogram data from the Helioseismic and Magnetic Imager (HMI). These measure the intensity of the random motions of magnetic elements and the state of turbulence of the magnetic field, respectively. The time changes of the non-thermal energy release in the corona was explored via histogram analysis of the non-thermal velocity, v {sub nt}, in order to highlight the largest values at each time, which may indicate an increase in energy release in the corona. We used the 10% upper range of the histogram of v {sub nt} (which we called V {sup upp} {sub nt}) of the coronal spectral line of Fe XII 195 A. A 2 day time interval was analyzed from HMI data, along with the EIS data for the same field of view. Our main findings are the following. (1) The magnetic turbulent diffusion coefficient, {eta}(t), precedes the upper range of the v {sub nt} with the time lag of approximately 2 hr and the cross-correlation coefficient of 0.76. (2) The power-law index, {alpha}, of the magnetic power spectrum precedes V {sup upp} {sub nt} with a time lag of approximately 3 hr and the cross-correlation coefficient of 0.5. The data show that the magnetic flux dispersal in the photosphere is relevant to non-thermal energy release dynamics in the above corona. The results are consistent with the nanoflare mechanism of the coronal heating, due to the time lags being consistent with the process of heating and cooling the loops heated by nanoflares.

79 ASTRONOMY AND ASTROPHYSICS↗

ABRUPT LONGITUDINAL MAGNETIC FIELD CHANGES AND ULTRAVIOLET EMISSIONS ACCOMPANYING SOLAR FLARES

We have used Transition Region and Coronal Explorer 1600 A images and Global Oscillation Network Group (GONG) magnetograms to compare ultraviolet (UV) emissions from the chromosphere to longitudinal magnetic field changes in the photosphere during four X-class solar flares. An abrupt, significant, and persistent change in the magnetic field occurred across more than 10 pixels in the GONG magnetograms for each flare. These magnetic changes lagged the GOES flare start times in all cases, showing that they were consequences and not causes of the flares. Ultraviolet emissions were spatially coincident with the field changes. The UV emissions tended to lag the GOES start times for the flares and led the changes in the magnetic field in all pixels except one. The UV emissions led the photospheric field changes by 4 minutes on average with the longest lead being 9 minutes; however, the UV emissions continued for tens of minutes, and more than an hour in some cases, after the field changes were complete. The observations are consistent with the picture in which an Alfven wave from the field reconnection site in the corona propagates field changes outward in all directions near the onset of the impulsive phase, including downward through the chromosphere and into the photosphere, causing the photospheric field changes, whereas the chromosphere emits in the UV in the form of flare kernels, ribbons, and sequential chromospheric brightenings during all phases of the flare.

79 ASTRONOMY AND ASTROPHYSICS↗

ON THE ROLE OF ROTATING SUNSPOTS IN THE ACTIVITY OF SOLAR ACTIVE REGION NOAA 11158

We study the role of rotating sunspots in relation to the evolution of various physical parameters characterizing the non-potentiality of the active region (AR) NOAA 11158 and its eruptive events using the magnetic field data from the Helioseismic and Magnetic Imager (HMI) and multi-wavelength observations from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory. From the evolutionary study of HMI intensity and AIA channels, it is observed that the AR consists of two major rotating sunspots, one connected to a flare-prone region and another with coronal mass ejection (CME). The constructed space-time intensity maps reveal that the sunspots exhibited peak rotation rates coinciding with the occurrence of major eruptive events. Further, temporal profiles of twist parameters, namely, average shear angle, {alpha}{sub av}, {alpha}{sub best}, derived from HMI vector magnetograms, and the rate of helicity injection, obtained from the horizontal flux motions of HMI line-of-sight magnetograms, correspond well with the rotational profile of the sunspot in the CME-prone region, giving predominant evidence of rotational motion causing magnetic non-potentiality. Moreover, the mean value of free energy from the virial theorem calculated at the photospheric level shows a clear step-down decrease at the onset time of the flares revealing unambiguous evidence of energy release intermittently that is stored by flux emergence and/or motions in pre-flare phases. Additionally, distribution of helicity injection is homogeneous in the CME-prone region while in the flare-prone region it is not and often changes sign. This study provides a clear picture that both proper and rotational motions of the observed fluxes played significant roles in enhancing the magnetic non-potentiality of the AR by injecting helicity, twisting the magnetic fields and thereby increasing the free energy, leading to favorable conditions for the observed transient activity.

79 ASTRONOMY AND ASTROPHYSICS↗

SYNOPTIC MAPPING OF CHROMOSPHERIC MAGNETIC FLUX

We used daily full-disk Ca II 854.2 nm magnetograms from the Synoptic Optical Long Term Investigations of the Sun (SOLIS) facility to study the chromospheric magnetic field from 2006 April through 2009 November. We determined and corrected previously unidentified zero offsets in the SOLIS magnetograms. By tracking the disk passages of stable unipolar regions, the measured net flux densities were found to systematically decrease from the disk center to the limb by a factor of about two. This decrease was modeled using a thin flux tube model with a difference in signal formation height between the center and limb sides. Comparison of photospheric and chromospheric observations shows that their differences are largely due to horizontal spreading of magnetic flux with increasing height. The north polar magnetic field decreased nearly linearly with time during our study period while the south polar field was nearly constant. We used the annual change in the viewing angle of the polar regions to estimate the radial and meridional components of the polar fields and found that the south polar fields were tilted away from the pole. Synoptic maps of the chromospheric radial flux density distribution were used as boundary conditions for extrapolation of the field from the chromosphere into the corona. A comparison of modeled and observed coronal hole boundaries and coronal streamer positions showed better agreement when using the chromospheric rather than the photospheric synoptic maps.

79 ASTRONOMY AND ASTROPHYSICS↗

SOLAR MAGNETIC ACTIVITY CYCLES, CORONAL POTENTIAL FIELD MODELS AND ERUPTION RATES

We study the evolution of the observed photospheric magnetic field and the modeled global coronal magnetic field during the past 3 1/2 solar activity cycles observed since the mid-1970s. We use synoptic magnetograms and extrapolated potential-field models based on longitudinal full-disk photospheric magnetograms from the National Solar Observatory's three magnetographs at Kitt Peak, the Synoptic Optical Long-term Investigations of the Sun vector spectro-magnetograph, the spectro-magnetograph and the 512-channel magnetograph instruments, and from Stanford University's Wilcox Solar Observatory. The associated multipole field components are used to study the dominant length scales and symmetries of the coronal field. Polar field changes are found to be well correlated with active fields over most of the period studied, except between 2003 and 2006 when the active fields did not produce significant polar field changes. Of the axisymmetric multipoles, only the dipole and octupole follow the poles whereas the higher orders follow the activity cycle. All non-axisymmetric multipole strengths are well correlated with the activity cycle. The tilt of the solar dipole is therefore almost entirely due to active-region fields. The axial dipole and octupole are the largest contributors to the global field except while the polar fields are reversing. This influence of the polar fields extends to modulating eruption rates. According to the Computer Aided CME Tracking, Solar Eruptive Event Detection System, and Nobeyama radioheliograph prominence eruption catalogs, the rate of solar eruptions is found to be systematically higher for active years between 2003 and 2012 than for those between 1997 and 2002. This behavior appears to be connected with the weakness of the late-cycle 23 polar fields as suggested by Luhmann. We see evidence that the process of cycle 24 field reversal is well advanced at both poles.

79 ASTRONOMY AND ASTROPHYSICS↗

FIRST SYNOPTIC MAPS OF PHOTOSPHERIC VECTOR MAGNETIC FIELD FROM SOLIS/VSM: NON-RADIAL MAGNETIC FIELDS AND HEMISPHERIC PATTERN OF HELICITY

We use daily full-disk vector magnetograms from Vector Spectromagnetograph on Synoptic Optical Long-term Investigations of the Sun system to synthesize the first Carrington maps of the photospheric vector magnetic field. We describe these maps and make a comparison of the observed radial field with the radial field estimate from line-of-sight magnetograms. Furthermore, we employ these maps to study the hemispheric pattern of current helicity density, H{sub c} , during the rising phase of solar cycle 24. The longitudinal average over the 23 consecutive solar rotations shows a clear signature of the hemispheric helicity rule, i.e., H{sub c} is predominantly negative in the north and positive in the south. Although our data include the early phase of cycle 24, there appears to be no evidence for a possible (systematic) reversal of the hemispheric helicity rule at the beginning of the cycle as predicted by some dynamo models. Furthermore, we compute the hemispheric pattern in active region latitudes (-30 Degree-Sign {<=} {theta} {<=} 30 Degree-Sign ) separately for weak (100 G < |B{sub r} | < 500 G) and strong (|B{sub r} | > 1000 G) radial magnetic fields. We find that while the current helicity of strong fields follows the well-known hemispheric rule (i.e., {theta} {center_dot} H{sub c} < 0), H{sub c} of weak fields exhibits an inverse hemispheric behavior (i.e., {theta} {center_dot} H{sub c} > 0), albeit with large statistical scatter. We discuss two plausible scenarios to explain the opposite hemispheric trend of helicity in weak and strong field regions.

79 ASTRONOMY AND ASTROPHYSICS↗

INTERPRETING ERUPTIVE BEHAVIOR IN NOAA AR 11158 VIA THE REGION'S MAGNETIC ENERGY AND RELATIVE-HELICITY BUDGETS

In previous works, we introduced a nonlinear force-free method that self-consistently calculates the instantaneous budgets of free magnetic energy and relative magnetic helicity in solar active regions (ARs). Calculation is expedient and practical, using only a single vector magnetogram per computation. We apply this method to a time series of 600 high-cadence vector magnetograms of the eruptive NOAA AR 11158 acquired by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory over a five-day observing interval. Besides testing our method extensively, we use it to interpret the dynamical evolution in the AR, including eruptions. We find that the AR builds large budgets of both free magnetic energy and relative magnetic helicity, sufficient to power many more eruptions than the ones it gave within the interval of interest. For each of these major eruptions, we find eruption-related decreases and subsequent free-energy and helicity budgets that are consistent with the observed eruption (flare and coronal mass ejection (CME)) sizes. In addition, we find that (1) evolution in the AR is consistent with the recently proposed (free) energy-(relative) helicity diagram of solar ARs, (2) eruption-related decreases occur before the flare and the projected CME-launch times, suggesting that CME progenitors precede flares, and (3) self terms of free energy and relative helicity most likely originate from respective mutual terms, following a progressive mutual-to-self conversion pattern that most likely stems from magnetic reconnection. This results in the non-ideal formation of increasingly helical pre-eruption structures and instigates further research on the triggering of solar eruptions with magnetic helicity firmly placed in the eruption cadre.

79 ASTRONOMY AND ASTROPHYSICS↗

Characteristics of the photospheric magnetic field associated with solar flare initiation

The physical environment governing the solar flare initiation is not fully understood, although there are significant efforts to address the relationship between magnetic non-potential parameters and early flare signatures. In this study, we attempt to characterize the flare initiation based on the processed Helioseismic and Magnetic Imager vector magnetograms, Atmospheric Imaging Assembly 1600 Å, and RHESSI hard X-ray observations. Three flare events, the M6.6 flare on 2011 February 13, the X2.2 flare on 2011 February 15, and the X2.1 flare on 2011 September 6, in two active regions AR 11158 and AR 11283 are investigated. We analyze the source field strength in the photosphere, which is defined as the magnitude of the observed magnetic field deviation from the potential field. It is found that one of the strong source field regions above the magnetic polarity inversion line well connects the initial bright kernels of two conjugate ribbons. The results imply that the distribution of the photospheric source field strength can be used to locate the initiation site of flaring loops regardless of the configuration of pre-flare magnetic fields or the evolution of active regions. Moreover, the field configuration in the strong source field regions tends to become more inclined after flares, which is consistent with the coronal implosion scenario. We also employ a fast method to derive the total current density from the photospheric vector magnetogram in the framework of force-free field. This method can provide fast estimation of photospheric current density within a reasonable accuracy without appealing for the more accurate calculation from a model extrapolation.

79 ASTRONOMY AND ASTROPHYSICS↗

Separable solutions of force-free spheres and applications to solar active regions

We present a systematic study of the force-free field equation for simple axisymmetric configurations in spherical geometry and apply it to the solar active regions. The condition of separability of solutions in the radial and angular variables leads to two classes of solutions: linear and nonlinear force-free fields (NLFF). We have studied these linear solutions and extended the nonlinear solutions for the radial power law index to the irreducible rational form n = p/q, which is allowed for all cases of odd p and cases of q > p for even p, where the poloidal flux ψ∝1/r{sup n} and the field B∝1/r {sup n+2}. We apply these solutions to simulate photospheric vector magnetograms obtained using the spectropolarimeter on board Hinode. The effectiveness of our search strategy is first demonstrated on test inputs of dipolar, axisymmetric, and nonaxisymmetric linear force-free fields. Using the best fit, we build three-dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent methods, which are in agreement. We have analyzed five magnetograms for AR 10930 spanning a period of three days during which two X-class flares occurred and found the free energy and relative helicity of the active region before and after the flare; our analysis indicates a peak in these quantities before the flare events, which is consistent with the other results. We also analyzed single-polarity regions AR 10923 and 10933, which showed very good fits to potential fields. This method provides useful reconstruction of NLFF and input fields for other numerical techniques.

79 ASTRONOMY AND ASTROPHYSICS↗

Statistical study of free magnetic energy and flare productivity of solar active regions

Photospheric vector magnetograms from the Helioseismic and Magnetic Imager on board the Solar Dynamic Observatory are utilized as the boundary conditions to extrapolate both nonlinear force-free and potential magnetic fields in solar corona. Based on the extrapolations, we are able to determine the free magnetic energy (FME) stored in active regions (ARs). Over 3000 vector magnetograms in 61 ARs were analyzed. We compare FME with the ARs' flare index (FI) and find that there is a weak correlation (<60%) between FME and FI. FME shows slightly improved flare predictability relative to the total unsigned magnetic flux of ARs in the following two aspects: (1) the flare productivity predicted by FME is higher than that predicted by magnetic flux and (2) the correlation between FI and FME is higher than that between FI and magnetic flux. However, this improvement is not significant enough to make a substantial difference in time-accumulated FI, rather than individual flare, predictions.

79 ASTRONOMY AND ASTROPHYSICS↗

Distribution of electric currents in sunspots from photosphere to corona

We present a study of two regular sunspots that exhibit nearly uniform twist from the photosphere to the corona. We derive the twist parameter in the corona and in the chromosphere by minimizing the difference between the extrapolated linear force-free field model field lines and the observed intensity structures in the extreme-ultraviolet images of the Sun. The chromospheric structures appear more twisted than the coronal structures by a factor of two. Further, we derive the vertical component of electric current density, j{sub z} , using vector magnetograms from the Hinode Solar Optical Telescope (SOT). The spatial distribution of j{sub z} has a zebra pattern of strong positive and negative values owing to the penumbral fibril structure resolved by Hinode/SOT. This zebra pattern is due to the derivative of the horizontal magnetic field across the thin fibrils; therefore, it is strong and masks weaker currents that might be present, for example, as a result of the twist of the sunspot. We decompose j{sub z} into the contribution due to the derivatives along and across the direction of the horizontal field, which follows the fibril orientation closely. The map of the tangential component has more distributed currents that are coherent with the chromospheric and coronal twisted structures. Moreover, it allows us to map and identify the direct and return currents in the sunspots. Finally, this decomposition of j{sub z} is general and can be applied to any vector magnetogram in order to better identify the weaker large-scale currents that are associated with coronal twisted/sheared structures.

79 ASTRONOMY AND ASTROPHYSICS↗